Lady Gaga walks into the room carrying a teacup. Nothing unusual there, you might think. But this isn't just any old teacup. Since an unfortunate incident a few weeks ago when it was briefly separated from its owner, the cup is, on the fame stakes, up there with Gaga herself. It might even have eclipsed her. I wouldn't be at all surprised if has its own book deal.

The saga began when Gaga took the teacup, a pretty purple number with flowers on the outside, on to Jonathan Ross's TV chat show. Hours later she was snapped by the paparazzi still carrying it. And the following evening she took the cup with her to a restaurant where, after a few cocktails, she accidentally left it behind. As it was subsequently reported, Gaga "kicked up a stink" and demanded that a taxi be sent back to the restaurant to retrieve it.

The following morning, Gaga's publicist, who was in the midst of spring-cleaning his bathroom, found himself fielding calls from showbiz journalists about the alleged hissy fit. Meanwhile, staff at her hotel were besieged by reporters looking for witnesses. On Monday the red-tops all carried articles on "Lady Gaga's Teacup Tantrum" and, inevitably, the "Storm in a Teacup". On Google, there were 50,000 related stories, including earnest blogs pondering the merits of the teacup as fashion accessory.

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For Gaga it's all grist to the mill in the weird world of pop stardom, a ridiculous moment of unpredictability in her otherwise regimented existence. While it's still unclear whether a tantrum actually took place, she certainly doesn't seem like the kind of person who would easily lose her cool. Despite spending her every waking hour in the eye of a publicity storm, she is a remarkably composed presence. "This is what I have chosen, or rather it has chosen me," she states huskily, settling down on a sofa. "This is what fame is."

We meet at a recording studio just off London's Oxford Street. Outside in the reception, Gaga's retinue of stylists, personal assistants, publicists, pluggers and managers sit staring at their Blackberrys, all plotting her next move. It's 10.30 in the morning. She has already put in appearances on BBC Breakfast news and Radio 1. Later she is due at Capital Radio where she will play songs in front of a select audience of fans. Then it's straight on a plane and back to her native New York where the whole merry-go-round will start over again.

With the record industry suffering from the combined effects of an internet revolution and global recession, there's a sense that Gaga is the goose that laid the golden egg. Apparently it's not enough that she has reached number one in 20 countries with her single "Poker Face", or been nominated for a Grammy for the song "Just Dance", or sold 1.9 million copies of her debut album, The Fame. The more impressive the statistics, it seems, the more intense the schedule.

But if it's hard work being an über-cool, multi-million-selling post-modern pop icon, Gaga isn't going to be caught grumbling. I spend a good 10 minutes trying to get her to admit that the interviews, the television appearances, the costume changes, the rictus smiles, are beginning to wear her down. After all, nobody spends years trying to make it as an artist in order to be quizzed about a teacup on breakfast telly, do they?

"You know, I have such an appreciation for where I am in my life because I've struggled and because I couldn't get signed, and because I couldn't get played on the radio," she says serenely. "There are times when it can be a lot to deal with but always when I get up in the morning I try to find that very joyful place that reminds me that I would die if someone took it all away. If someone did that I wouldn't be a person anymore."

Gaga is 23, though with her white hair, radioactive tan and pale-pink frosted lipstick, she appears oddly ageless, as if she's been cryogenically frozen and is still in the process of defrosting. Today she is wearing a white shoulder-padded jacket, black leggings, vertiginous platform sandals and a huge harlequin-style hat. And let's not forget her trademark: the false eyelashes, so heavy-duty that her eyes don't seem to open properly.

Gaga is notorious for her outré appearance, of course. On the Brit Awards she performed with the Pet Shop Boys in a porcelain bikini, while on her recent tour she appeared on stage in a clear plastic bubble dress. On Friday Night With Jonathan Ross she rocked a toga-style number made entirely of red vinyl Post-It notes. Gaga claims she would look like this whether she was famous or not. The eyelashes hardly ever come off; she even sleeps in them. "Whenever I have a lover I leave them in their apartment on the pillow," she coos. "Kind of like a keepsake."

Gaga, who took her name from the Queen song "Radio Gaga", has a steeliness and ambition that is at once impressive and terrifying. Not for nothing has she been compared to that other hugely successful mistress of provocation, Madonna. But the difference is that Gaga has created a persona that is so strong it has overwhelmed her entirely.

"Right now the only thing that I am concerned with in my life is being an artist," she states. "I had to suppress it for so many years in high school because I was made fun of but now I'm completely insulated in my box of insanity and I can do whatever I like."

As Gaga sees it, she's is a living subversion of the modern pop diva. She talks loftily about her art. Her stage designs are not sets, they're installations. She is not a pop star, she's a performance artist. Her idols are those who have blurred the boundaries between popular art forms, from Piet Mondrian and Andy Warhol to David Bowie and David LaChapelle.

"My ideas about fame and art are not brand new," she says. "We could watch Paris is Burning [Jennie Livingston's 1990 documentary about New York drag artists], we could read The Warhol Diaries, we could go to a party in New York in 1973 and these same things would be being talked about. I guess you could say that I'm a bit of a Warholian copycat. Some people say everything [in music and fashion] has been done before, and to an extent they are right. I think the trick is to honour your vision and reference and put together things that have never been put together before. I like to be unpredictable, and I think it's very unpredictable to promote pop music as a highbrow medium."

Gaga's articulation of her position is certainly rare. This is not, one imagines, the kind of conversation you would get from Britney Spears. But listening to her album The Fame, a disco-tinged, sexually charged critique of modern pop culture, it's hard to distinguish Lady Gaga's sound from that of her chart-friendly contemporaries, from Pink to Christina Aguilera. Her songs are undoubtedly catchy but there is a gulf between how she perceives them and what the rest of us actually hear.

What really sets Gaga apart from her contemporaries is that she is responsible for every part of her act. She directs her own shows, designs the sets, chooses the clothes and writes the songs. She never, ever lip-synchs. Despite her coterie of creatives back home, whom she calls the "Haus of Gaga", she retains complete control over everything. If Gaga has spent the past two years fending off suggestions that she is nothing more than the product of a record-company brainstorm, she is not offended: "It's exactly the kind of discussion that I want to start."

But in deconstructing the cliché that is the modern pop star while enjoying all the benefits that the role affords, isn't there a sense, I ask, that she has had her cake and eaten it?

"It's not parody, it's commentary," Gaga replies coolly. "To use the words 'have your cake and eat it' implies something devious. For me, I just think I'm very good at what I do."

Still, there are moments when Gaga's conceptualising can be a little hard to follow. "I don't ever want to be grounded in reality," she explains when I ask if her persona is really sustainable 24 hours a day. "In my show I announce, 'People say Lady Gaga is a lie, and they are right. I am a lie. And every day I kill to make it true.' It's the dream of my vision, it's the lie that I tell, whether it's an umbrella or it's a hat or it's the way that I shape my lipstick. And then eventually it becomes a reality. My hair bow was a lie and now it's true."

Quite. What is clear is that Gaga is a very modern artist, not just in her sound and look, but also in how she courts the media. If her every moment is a performance, then the paparazzi are her enablers, instantly beaming images of her around the world. Since her rise to fame, barely a day has passed without Gaga appearing in the papers buying fish'n'chips in a fluorescent leotard or stepping out for the night in her PVC catsuit. Gaga lets out a little sigh at the mention of the photographers who are currently loitering around the front door. Put on a baggy sweater and you could lose them, I say. She looks at me aghast.

"That's a very dangerous precedent, and it's not fair to my fans. They don't want to see me that way just like I don't want to see Bowie in a tracksuit. He never let anyone see him that way. The outlet for my work is not just the music and the videos, it's every breathing moment of my life. I'm always saying something about art and music and fame. That's why you don't ever catch me in sweatpants."

The eldest daughter of an internet entrepreneur and his business-partner wife, Gaga – born Stefani Joanne Angelina Germanotta – was brought up in New York. She was a proficient pianist at five; by the time she was 11 she was all set to go to the prestigious Julliard School in Manhattan, though she decided it wasn't for her. Instead she went to the Convent of the Sacred Heart, also attended by Paris and Nicky Hilton. She was, she says, "focused, determined. I was always in a band, or in a musical. I didn't really fit in but I had friends because I'm a nice girl and fun to party with."

From the age of 14 Gaga was playing at New York clubs, which I note must be illegal until she tells me that she was chaperoned by her mother. Really?

"These were jazz bars not sex clubs," she explains. "They would have open mic nights so my mother would take me along and say, 'My daughter's very young but she's very talented. I'll sit with her as she plays.' "

An A-grade student, Gaga got a place at NYU's Tisch School of Arts a year early at 17. But she abandoned it after one year in favour of what she saw as a more honest education, plying her trade on the Lower East Side club scene. Immersing herself in the seamier side of life, she dabbled in drugs and worked as a burlesque singer. During performances she would frequently set light to cans of hairspray. But she never stopped playing gigs.

"There was this one night where I had had a couple of drinks," she remembers. "I had new material and I had on this amazing outfit. So I sat down, cleared my throat and waited for everyone to go quiet. It was a bunch of frat kids from the West Village and I couldn't get them to shut up. I didn't want to start singing while they were talking, so I got undressed. There I was sitting at the piano in my underwear. So they shut up."

It was then, she says, that Lady Gaga was born. "That's when I made a real decision about the kind of pop artist that I wanted to be. Because it was a performance art moment there and then." She pauses and thinks. "You see, you can write about it now and it will sound ridiculous. But the truth is that unless you were in the audience in that very spontaneous moment, it doesn't mean anything. It's, like, she took her clothes off, so sex sells, right? But in the context of that moment, in that neighbourhood, in front of that audience, I was doing something radical."

Gaga's pop ascendancy began in earnest a year later when the R&B star Akon heard her music and asked her to help him write some songs.

"I was like the weird girl who dressed like a zoo animal, the trash glamour in a roomful of urban hip-hop cats," she smiles. "They'd be, like, 'Gaga, what do you think of this lyric?' and I'd twist it all up and all of a sudden it was edgy."

Overnight she became a pop writer for hire, producing songs for the likes of Britney Spears, the Pussycat Dolls and New Kids on the Block. Was it frustrating, writing for other people when were trying to make it as a singer yourself?

"Honestly no, I loved it," she replies. "I got real joy from hearing Britney Spears sing my melodies. As much as I can sit here and talk about art, there's still something quite remarkable about writing a song when you're 20 and hearing a pop superstar sing it."

All the while, Gaga was putting together her own album. She had been signed by Def Jam records but was dropped shortly after. Next she signed with Interscope, though this time it was the radio stations that were proving resistant.

"They would say, 'This is too racy, too dance-oriented, too underground. It's not marketable.' And I would say, 'My name is Lady Gaga, I've been on the music scene for years, and I'm telling you, this is what's next.' And look," – she gestures grandly around her – "I was right."

It's getting on for midday and our time is up. Gaga is due a half-hour break before going to Capital Radio. Time for a power nap, I ask? "No, I'm going to work on my songs," she says sunnily and, picking up her teacup, teeters out of the room.

When it's time to leave, Gaga puts on her shades, opens her umbrella and, flanked by her security men, climbs into a people carrier. The publicist and I hop in a car behind. As we set off a photographer appears from nowhere on a scooter, and dices with death as he tries to stay close. There are at least 20 more paparazzi waiting outside the radio station, all of whom look desolate when the people carrier disappears, Gaga and all, into the tradesman's entrance.

Ten minutes later, Gaga is settled at the keyboards in the studio. The audience files in and the retinue return to their Blackberrys. Gaga performs three songs, including a classical version of her hit "Poker Face", during which she hoists a leg on to the keyboard and plays piano with her heels, and "Paparazzi" which she dedicates to "those buggers outside". It's a witty, stylish performance that easily proves her worth as a performer. But watching as the mostly teenage fans mouth along to every word, I wonder to what extent the performance-art concept is really appreciated.

"I have found that my work has to be both deep and shallow," said Gaga when I put this to her earlier. "All of my songs have meaning, all of my clothing has iconography buried into it. But by the same token, it's just as special if you look at it in its shallowest form. A quick moment of melody, a beautiful dress. People think, 'Gaga's so sweet', or 'Gaga sucks'. The point is that it's memorable. For commercial art to be taken seriously as fine art is a very unusual and difficult task. I think that a lot of people don't get it and a lot of people don't know what to make of me. And, you know what? I'm OK with that."

Lady Gaga's album, 'The Fame', is out now on Polydor. Her new single, 'Paparazzi', will be released in June. She will be playing Manchester Academy on 29 June, Oasis in Swindon on 6 July and 02 Academy in Brixton on 14 July. For tickets call 08444 775775