Music: Scarlett Johansson finds her voice

UNDER April skies, on a stage in a polo field in the Californian desert, I am watching two East Kilbride brothers in their forties bury years of enmity, distance and boozing.

They have some help: reliable backing musicians, a tasty fee from the concert promoters, and – lurking stage-left, under a wide-brimmed fedora, hanging on to a microphone as if her life depended on it – a vacationing Hollywood princess.

It is spring 2007 and Jim and William Reid are launching the comeback of the Jesus And Mary Chain with a show at the Coachella Festival. It's a reunion many thought would never happen. The fact that the infamously argy-bargy siblings have enlisted Scarlett Johansson to sing 'Just Like Honey' only adds to the oddness, the specialness, of the occasion. "Is that really her?" is the buzz going round Coachella, swiftly followed by: "Hey, she's not bad." That karaoke scene in Lost In Translation – in which Johansson sang the Pretenders' 'Brass In Pocket' and Bill Murray did his surprisingly rockin' version of Roxy Music's 'More Than This' – was no fluke.

"It was surreal," the actress recalls of her Coachella guest slot with the brothers grim. "I had been such a fan and to meet them backstage was like: 'OK, here I am at your disposal, I guess.' I didn't know what parts I was supposed to sing, but I knew how fantastic I sounded in the shower when I sing their songs."

One year on, Johansson is in London, in support of an even more surprising endeavour: her debut album, Anywhere I Lay My Head, a collection of covers of Tom Waits songs. More surprising still: it's great. We're not talking Minnie Driver doing her Sheryl Crow-meets-Katie Melua posho-singer-songwriter schtick. Banish thoughts of Bruce Willis trying to find some soul. We're not even in the realms of Billy Bob Thornton turning out decent albums of gravelly Americana. Anywhere I Lay My Head is no dilettante-ish indulgence, nor an easy option cash-in, but an admirably wayward art project.

Johansson says she was approached to make an album by record label Rhino, whose people had been impressed by her contribution to a 2006 charity record, Unexpected Dreams: Songs From The Stars. "I have so many friends that would kill to have that opportunity, so I couldn't pass it up," the 23-year-old New Yorker says as she settles into a chair in a basement room of a private members' club in London's Covent Garden. "But I wasn't quite sure what to do with the opportunity. Initially I thought I would do a bunch of standards because I always loved Rosemary Clooney, Billie Holiday and Ella Fitzgerald, and other female jazz vocalists. I wanted to include this Tom Waits song, 'Never Talk To Strangers', and I thought I would put more Tom Waits songs on. Eventually it became all Tom Waits songs."

Another breakthrough was the hiring of Dave Sitek, the brains behind New York art-rock outfit TV On The Radio, who has just been named No 1 in the NME's Future 50 list. He and Johansson discussed what kind of record they wanted: in his inimitable phrase, they decided to "make it sound like we drank a lot of cough medicine and saw Tinkerbell".

Hear the 11 tracks on Anywhere I Lay My Head – there's also a Sitek/Johansson composition, the psychedelic acoustica of 'Song For Jo' – and you get what he means. It's a dreamy, atmospheric collection, recalling This Mortal Coil or Cocteau Twins. On the title track (from Waits' Rain Dogs album) and 'Town With No Cheer' (from Swordfishtrombones), virtue has been made of the limitations of Johansson's deep voice: it's one more sound in a wash of Spector-esque drums, fuzz guitar (courtesy of another NYC hipster, Yeah Yeah Yeahs' Nick Zinner) and ghostly bleeps.

Another surprise guest is David Bowie, who came to the recording studio in Louisiana to sing backing vocals on 'Falling Down' (from Big Time) and 'Fannin Street' (from the 'Bawlers' half of Waits' Orphans collection). He's a big TV On The Radio fan, plus he and Johansson knew each other from working on The Prestige.

"I realised I was a sexual being from listening to Bowie as a young adolescent," beams Johansson – who, even when she's dressed down in an indie-cardie, manages to look sultry. "I had met him at a show one time and then I was at a dinner party and he was sitting at my table and he knew I was working with Dave… I probably said something jokingly like: 'If you have any free time then come down.'

"And then one day on a fated afternoon this past summer I was in Spain and I get this call from Dave saying: 'You will never believe who's in the studio recording right now.' It was magical. I wanted to die! I was like, the one time I can't believe I'm not there. When I got the tracks I almost fainted."

Anywhere I Lay My Head is a hypnotic, defiantly contrary and borderline unsettling record. Robert Plant and Alison Krauss do a version of Waits's 'Trampled Rose' on their superlative Raising Sand album. Intriguingly, this Hollywood star has covered the rough-voiced boho-barfly with more edge than the former Led Zeppelin frontman.

"I always loved to sing," Johansson says. "I started acting because I wanted to be in musical theatre. I took vocal lessons when I was young, and then hit puberty and got self-conscious about being on stage. I always had the love for singing but just didn't want to be a showman any more. Some people may like or dislike how I sound but I felt confident that I could interpret these songs strongly enough. That's why I had the crazy idea to believe I could do it in the first place."

Any word on what Waits thinks of the record? "I made sure he was going to be into it, and we weren't just taking his material and running with it," she says, adding that she hasn't managed to speak with him directly. "He gave us the thumbs-up. So I was like: 'OK, now I can proceed.' And from what I've heard I think he's very pleased with the result." Next up, how about Ms Johansson's take on the Mary Chain's Psychocandy?

Scarlett Johansson, Anywhere I Lay My Head is out May 19 on Warner