Patti Smith was 20 years old when she fell in love with Rimbaud. By the time she read Illuminations, the poet had been dead for over 70 years but that did not seem to matter. "My unrequited love for him was as real to me as anything I had experienced," Smith writes in an early chapter of Just Kids. The "hard-edged, illiterate" women with whom she worked in a factory were suspicious of Smith's predilection for foreign literature and accused her of being a communist. "I was harassed in his name," she continues, breathlessly. "It was for him that I wrote and dreamed his hands had chiselled a manual of heaven and I held them fast."

Much of the first half of Just Kids is dominated by Rimbaud and countless mentions of the other important men in Smith's life, most of whom shared the principal attributes of being French, dead and terribly artistic. Baudelaire, Cocteau and Genet all merit frequent references and we are left in no doubt that Smith has her mind on higher things than factory work. After giving up a child for adoption she buys a one-way ticket to New York and disappears into an esoteric mist of artistic pretension. The friends she is hoping to stay with never show up so she is forced to take to the streets. When a tramp asks her, "What's your situation?" she replies, "On Earth or in the universe?"

Fortunately both Smith and the book are saved from imploding with self-satisfaction by a chance encounter with a green-eyed boy called ­Robert ­Mapplethorpe. In Mapplethorpe, Smith finds her spiritual twin, a man as obsessed with artistic creation as she is, "a lover and a friend to create with, side by side. To be loyal, yet be free".

For the next 12 years, against the vivid backdrop of 1970s New York, Mapplethorpe and Smith would live together, support each other and share jointly in their burgeoning success as artists  Mapplethorpe as a photographer, Smith as a poet turned rock'n'roll singer who would become the mother of punk. "As if it was the most natural thing in the world we stayed together, not leaving each other's side save to go to work," she writes. "Nothing was spoken, it was just mutually understood."

They slept together too, and although Mapplethorpe later admitted he was gay, his relationship with Smith remained intimately cocooned from the outside world. As the title suggests, their lasting friendship was defined more than anything by its innocence and purity  two qualities that course through Smith's ­lyrical prose.

The relationship with Mapplethorpe infuses her writing with a necessary human warmth. The knowing references become less frequent and she concentrates instead on crafting a moving and delicately handled dual memoir, a love letter to the man who became her real-life Rimbaud. "To me, Robert and I were irrevocably entwined," Smith writes. "We played similar games, declared the most obscure objects treasure, and often puzzled friends and acquaintances by our indefinable devotion."

Living in a succession of squalid New York apartments, spending what little money they had on art supplies and surviving on day-old doughnuts and lettuce soup, both Mapplethorpe and Smith took their first tentative steps towards becoming the artists they so desired to be. Mapplethorpe, always the more focused and ambitious of the two, started to make collages by ripping photographs from male pornographic magazines. To save money, Smith suggested he take his own pictures. Some of the early portraits are reproduced here, Smith's gaunt elegance and dense-eyed gaze staring out of the pages in black and white. It was Mapplethorpe who took the iconic shot of Smith  at once both cocky and fragile  for the cover of her first album, Horses. Later, her husband, the guitarist Fred "Sonic" Smith, remarks, tellingly, that "all his photographs of you look like him".

Although both Smith and Mapplethorpe eventually went their separate ways  she got married and had two children; he embarked on a long-term relationship with the collector Sam Wagstaff  their spiritual closeness remained. When she discovers that Mapplethorpe is dying of Aids in 1989, Smith writes with brutal poignancy that "every fear I had once harboured seemed to materialise with the suddenness of a bright sail bursting into flames". Shortly before his death, when Mapplethorpe ruefully comments that they never had a family, Smith responds: "Our work was our children."

And it is true that in many ways, Just Kids is a compassionate portrait of an unconventional marriage; an intimacy forged through a shared artistic vision. In both the tenderness of her expression and the beauty of her prose, there is no doubt that Patti Smith has given us a fitting memorial to her lost love and to the art they created together.

Patti Smith has a mythic imagination. As a young, desperately poor poet from southern New Jersey, she headed to New York to seek her fortune, nothing in her purse. Her mother had assumed she would follow her into waitressing. But Patti, though practical and a survivor, had her sights set not on slinging hash but on searching for immortality and beauty and magic. She already recognised a divine succession of poets  Blake, Baudelaire, Rimbaud, Genet and the Beats  and she wanted to join them. She was creative and liked to write, read and draw. Eventually, she became the renaissance woman of the punks, a great rock singer and composer  but before that she had to fashion her look, her personality and her verse.

And survive. She had no real friends when she arrived in New York, just a few names, and no job prospects. But it was July 1967, she was not yet 21, and other drifters and hippies helped her find food and shelter. Eventually, she got a job working in a bookshop, she met Robert Mapplethorpe, who was the same age and just as poor, and they took a Brooklyn apartment together. They each collected little ­talismanic objects and set great store by the way they dressed; both had an ­innate and highly original sense of ­personal style. And he was fiercely ­ambitious and coveted artistic success.

In her careful, sometimes painful self-sculpting, Smith had found an inspired and equally determined collaborator in Mapplethorpe. As she says in this memoir, which is so full of memorable sentences: "We were both praying for Robert's soul, he to sell it and I to save it." (Robert's theme song was the Stones' "Sympathy for the Devil".)

Patti and Robert were both born in 1946 and both were raised by poor parents, she in Germantown, Pennsylvania and then New Jersey, he by a Catholic family on Long Island. Like all ­lovers, they told endless ­stories to each other about their childhoods: "We used to laugh at our small selves, saying that I was a bad girl ­trying to be good and that he was a good boy trying to be bad." They both succeeded. As a child he'd been a ­mama's boy and had made necklaces for his mother, but later, as an adult, he identified himself in the public mind through his photographs with pain and blood and exotic sexual ­practices, and even with something as seemingly transgressive (but actually innocent) as pictures of child ­nudity. She had held factory jobs in New Jersey, where the other workers accused her of being a communist ­because she was reading a bilingual edition of Rimbaud's Illuminations. She'd given birth out of wedlock, as we used to say, to a child she'd had to put up for adoption. Later, when she lived with Mapplethorpe in Brooklyn, she turned herself into a ­disciplined poet and breadwinner. For a long spell she supported the skinny, charismatic Mapplethorpe, who at the time was making "altars" of found objects somewhat in the manner of the American surrealist Joseph Cornell. He discovered photography only later, but once he settled on it as a career he was tenacious and highly tactical in plotting his rise in the world.

She obviously has a great gift for ­appreciation, though in her case that should not imply a lack of discrimination. It seems that from the very beginning she was alert to influences that would help her to explore and to firm up her peculiar sensibility, which was at once edgy and lyrical, both demotic and hieratic. She was more relaxed about their ability to survive; Robert was much more ­anxious about money. She was primarily interested in sniffing out people with talent, not as a careerist but ­always out of respect for their artistry. Mapplethorpe had his eye on the main chance.

In those days, before the internet and Google, it was difficult for ­working people to put their hands on books and information. All these years later, Smith still remembers the few art books she possessed and that she would consult again and again, just as she remembers their few records and books of verse. And she recalls in vivid detail her first encounters with William Burroughs and Allen Ginsberg, saints in her pantheon of great artists. For her, many sites in New York were ­sacred: "It was exciting just to stand in front of the hallowed ground of Birdland that had been blessed by John Coltrane, or the Five Spot on St Mark's Place where Billie Holiday used to sing, where Eric Dolphy and Ornette Coleman opened the field of jazz like human can openers." Robert's idols were visual artists, though very cerebral ones  ­Duchamp and Warhol. Patti indulged in long introspective bouts, but she learned from Robert just to get on with it and forge ahead in her work  a trajectory that for her was always God-centred, doing drawings "that magnified His motion".

While I was working on my biography of Jean Genet in the late 1980s and early 1990s in Paris, I would receive phone calls from Patti in Detroit. I'd never met her and was introduced to her only many years later in front of the Metropolitan Opera in New York, but she was calling me to encourage me to persevere and to finish this ­onerous seven-year task. She knew how ­devoted I had been to Mapplethorpe, with whom I'd collaborated on journalistic stories about Truman Capote and William Burroughs (I also wrote an ­essay for one of his first ­gallery shows, in Amsterdam). She told me that she and Robert (who had recently died of Aids) used to read Genet out loud to each other when they lived together. When my bio­graphy finally came out, Patti was staging her big comeback with a free concert in New York's Central Park; she told her audience that they must all go out to buy my Genet book.

This genuine devotion to her private artistic saints and to her old friends characterises the entire book. It is her own Lives of the Saints, and it is thoroughly imbued with faith in her own artistic mission.

Her love affair with Mapplethorpe, to be sure, had its painful moments, ­especially as they were both discovering that he was gay. Although they had been sexually intimate for several years, he began to pick up extra money as a rent boy. Jim Carroll, a friend who went on to be a punk musician and the author of the autobiographical The Basketball Diaries, was also hustling, in his case to support his heroin habit. When Mapplethorpe asked him how he could be certain he wasn't gay, ­Carroll said he'd never done it without being paid  which was not the case with Mapplethorpe. Before long, Robert had a handsome young lover and eventually a much older and even more handsome lover, Sam Wagstaff, a rich art collector who launched his career.

What Patti found even more difficult to accept than Robert's homosexuality was his social ascent. She could understand his love for men, but in order for her to spend time with his new, rich friends she would have had to change her ways.

Just Kids should interest any reader who wants to know how an artistic ­career can be launched. Smith gave a carefully staged and prepared poetry reading at St Mark's in New York that won her lots of attention and publication  and even the offer of a record contract. She began to work as a music journalist for Crawdaddy! and Rolling Stone. She begged the editor of Rolling Stone to let her write a piece on Lotte Lenya, Kurt Weill's wife, muse and ­favourite singer; when Patti handed in her article, the editor said "that although I talked like a truck driver, I had written an elegant piece". She had an affair with Carroll and with Sam Shepard, with whom she wrote a play.

Her transition to musician seems, in this account, to have been disconcertingly easy. She bought a guitar and soon knew how to play it. She turned some of her poems into songs. She put together a band  and before long she was a megastar touring the world. Mapplethorpe produced a portrait of her that undoubtedly helped to ­cement her image; with her gift for phrase-making, Patti writes: "Robert was ­concerned with how to make the ­photograph, and I with how to be the photograph." Suddenly, Robert was showing photos in galleries attended by "a ­perfect New York City mix of leather boys, drag queens, socialites, rock and roll kids and art collectors".

Like that art opening, this book brings together all the elements that made New York so exciting in the 1970s  the danger and poverty, the artistic seriousness and optimism, the sense that one was still connected to a whole history of great artists in the past. This was a small community that was carefully observed by the media; it also flourished at the moment when New York was becoming the cultural capital of the western world.

Edmund White's latest book is City Boy: My Life in New York During the 1960s and '70s.