The history of popular music has a fat chapter of hard luck stories — the artists who through bad advice, bad timing or bad drugs never reached their potential. Terence Trent D’Arby managed to sabotage a potentially stellar career; the Stone Roses imploded after one seminal album; for every Pink Floyd, there is a Syd Barrett.

All of which makes the second coming of Madeleine Peyroux the more remarkable. In 1996 this one-time busker was plucked from the streets to make a haunting album, Dreamland, that sold almost 250,000 copies. You probably won’t have heard of her — there was a sole British show — but she delighted the critics. Time magazine declared the album “the most exciting, involving vocal performance by a new singer this year”. Its mix of country, blues and jazz, sung in a sexy, soulful voice, predated Norah Jones’s 20-million-selling Come Away with Me by six years. John Major has recently come out as a fan, but don’t let that worry you.

However, despite high-profile touring, a second album never emerged. For the 23-year-old who grew up in Paris and Brooklyn, sudden success was all too much. Recordings were scrapped, with Peyroux blaming the weight of expectations and “vocal problems”. Soon she was back playing small clubs, street corners, even weddings. That is until this year: for Peyroux has gained that rarest thing in pop, a second chance. Once again a major label has taken her under its wing and the result is Careless Love, a CD on which her smoke-and-honey vocals — often compared to Billie Holiday — caress songs by Bessie Smith, Bob Dylan and Hank Williams. In a year that has seen the public bombarded with jazz-age nostalgia, from Clare Teal to Jamie Cullum (even the wretched Rod Stewart is at it), Careless Love is the year’s most beguiling and credible take on songs past.

Sitting in a hotel during a brief visit to the London Jazz Festival, Peyroux, who has the gamine look of the waif troubadour, seems unfazed by the way fame and fortune appeared to slip from her grasp (especially if you measure these things by the frenzied wannabes who populate The X Factor). “Maybe I seem not too bothered because a lot of things have gone right recently,” she says in her slow, thoughtful delivery. “I never saw success as inevitable. I felt that Dreamland’s reception might have been a fluke. I’ve never been quite sure of myself. A lot of people are great performers and they don’t survive by their craft, so why me?”

Nevertheless, she admits that the failure of the second album was wounding. “We went into the studio with a big producer, a big budget and we had musicians flown in and we had arrangements. We had all these wonderful ingredients . . . but we didn’t have the recipe.

“Maybe it was tough but maybe I’m not the kind of person who needs fame so much,” says Peyroux, the daughter of bohemian parents, who left home in Paris in her late teens to busk around the Continent.

With no release pending, her company, Atlantic, told Peyroux not to work. “That was the hardest part. There was a contractual understanding that I was to put a stop to any touring.”

She went for treatment at a vocal hospital in Nashville and stayed. “I began looking for other work, but the best-paid work I could get was singing. So I sang blues in a sports bar five nights a week. It was fun, it was comfortable.” Eventually wanderlust took over and she travelled around the States and returned to busking.

“It’s like riding a bicycle, it’s something you can never forget. To me it’s the warmest environment to work in. There are so many people who come up and say, ‘I’ve had a terrible day and you’ve changed my life just by singing for five minutes.’ After Dreamland I toured supporting Sarah McLachlan doing big shows — 5,000 people — but those little exchanges out busking have meant more to me.”

Peyroux’s progress could have been very different. The singer, now 31, has long been friendly with Jesse Harris, writer of the award-winning Don’t Know Why for Norah Jones. He has co-written a tune for Peyroux’s new album and Harris songs that appeared on Come Away with Me were originally offered to Peyroux. So in a parallel universe is it Peyroux who has sold 20 million?

She laughs. “She is different from me, more of an R&B singer. She’s very good at songs across the board but hers was more of a pop record, less jazz than me.”

It was Peyroux’s original producer, Yves Beauvais, who brought her back into the studio to make Careless Love. This time she is better prepared to capitalise on another set of glowing reviews. “I’m older and have a more relaxed perspective. I’m really happy to be working at this level again.” She grins. “It’s all being taken very seriously.”

Careless Love is released by Universal