Last week Jamie Cullum slipped on a tux, mussed up his hair and set off for a private party at St James's Palace, where he sat behind a grand piano and sang Sinatra's In the Wee Small Hours of the Morning to the Queen.

"She told me it was magical," he says, after bouncing into a brasserie near his Paddington flat, his mid-morning chirpiness belying a night spent drinking vodka with his mates. "I've never been into royalty," he adds, "but I was, like, bowing and saying, 'Hello, Your Majesty, please don't cut off my head.' It's all been a bit overwhelming."

In April, Universal Jazz beat Sony to sign Cullum for £1 million. The news sparked a media frenzy. Was the 5ft 5in, Wiltshire-raised 23-year-old worth the hype? When he sang, sighed and emoted his way through You're Nobody til Somebody Loves You on Michael Parkinson's BBC1 show shortly afterwards, the answer appeared to be yes. Online superstore Amazon ran out of his second CD, Pointless Nostalgic (released on the independent Candid label), the next day (He produced his first album, Heard It All Before, himself, in 2000).

As the story gathered momentum, Cullum's wide-ranging vocals and varied repertoire - classics from the Great American Songbook; jazz-inflected Radiohead and Nirvana covers; clever, catchy originals with ironic titles such as I Want to Be a Popstar - had him hailed as an antidote to the plague of manufactured pop. His boy-band glamour helped. Cullum is tipped to bring a new, young audience to jazz, a genre usually more associated with chin-stroking than underwear-throwing.

"I want to make jazz more accessible," says Cullum, whose sideline project, an unreconstructed rock band called Taxi, have supported Paul Weller and Toploader. "OK, I'm not pushing boundaries, but I am mixing things up a bit."

His as-yet-unnamed third album, due in September, will feature a version of Jimi Hendrix's The Wind Cries Mary, alongside more jazz standards and new material co-written with his session musician brother, Ben, 28.

Not all jazz fans are enamoured with Cullum's mix-and-match approach. Some purists argue that he is helping to make jazz fair game for pop Svengalis who have run out of ideas. But, says Cullum, the important thing is deciding what is jazz and what isn't. "Take Norah Jones's stuff. It's nice, but it's nice acoustic music. It's not jazz."

What, then, of Robbie Williams, whose adventures in swing have, like Cullum's, repackaged the Rat Pack for a new generation? And what of the criticisms that the effervescent Cullum is, like Williams, more entertainer than jazzman?

Cullum's shoulders sag. "What Robbie Williams did for the music was great," he says carefully, "but he didn't have the right voice. I know people who worked on that concert [Swing When You're Winning]; they had to change the keys for Robbie on those songs. We're not similar."

Showmanship matters, nonetheless. "Jazz gigs can be off-putting when you're watching guys just doing their thing and not saying anything. I know you're meant to be engaging with the music, but it helps if you're being entertained as well."

Doubters who accuse him of style over substance - the jazz old guard, mostly - are invited to come and see him play. A veteran of more than 1,000 live gigs, the largely self-taught Cullum has managed to win over even the most resistant British critics with his technique and taste.

His forthcoming residency at the legendary Algonquin Oak Room in New York in October is a first for a European jazz act, and he makes his debut at London's equally venerable Ronnie Scott's on 3 August. "Those gigs," he says, "are dreams come true. Way better than any record deal."

A penchant for showing off is in his genes. His parents - John, a second-generation Palestinian and Yvonne, second-generation Burmese - met when working at Ford Motors in Essex, and performed in a guitar-and-vocals pub act for much of Cullum's youth. "They did it for fun, for pocket money," he says.

His childhood in rural Hullavington, about 20 miles from Bristol, was "happy, normal, unremarkable". Both he and Ben were introduced to old-school jazz and rock via the family record collection. He taught himself guitar and piano and dabbled in acting at school - he was a reluctant Oliver in Oliver! ("I was small and I could sing") - before heading off to Paris in his gap year. There he read Kerouac, played the piano bars of the Left Bank and fell madly in love.

It didn't last. Cullum is cagey about what happened, but he got his heart broken. Careerwise, this wasn't such a bad thing; his covers of, say, Gershwin's I Can't Get Started are imbued with the requisite, if naive, world-weariness (he is currently seeing a "slightly older woman", but won't go into details). Back in the UK, he studied film and drama at Reading University (he is particularly proud of a self-produced short featuring a kleptomaniac courier and a depressed bloke on a hill) and played in hip hop, rock and jazz bands.

Musical or otherwise, Cullum's tastes have never been confined solely to jazz. With influences ranging from Thelonious Monk and Woody Allen to the writer Paul Auster and the Japanese film director Takeshi Kitano, he prefers to view everything he does as part of a whole. It's just that jazz is where his heart is.

"Even Prince William is getting into real music," he says. "He told me how great it was that people are listening to instruments again. He also said he likes techno when he feels like dancing and I agreed. Our views were pretty much the same, actually." Cullum stops and blinks, digesting the events of the past few months.

"You know, I took the Queen's place mat off her table afterwards," he confesses. "It's got the running order with my name on it. I don't care how uncool it is, I'm having it framed." He flashes a smile. "And then I'm sticking it next to the photo of me and Parkinson, up on my goddamn wall."

Jamie Cullum plays at the Cafe de Paris, W1, tonight, Barfly at the Monarch, NW1, on 1 July and Ronnie Scott's, W1, on 3 August.