Like Lee Anne, they all have stage names - Donovan, Sky, Taylor - and earn big money, as much as dollars 50,000 ( pounds 33,000) a year. Lee Anne, in her early twenties, is a tough, pretty British girl from Cardiff with auburn hair and a great body. She is thrilled with the life because, as she says, 'I earn all that money]'
In the VIP Lounge at Cabaret Royale, I ask Lee Anne to show me how she works a client, how she performs a 'table dance', the mainstay of the clubs. Along with dozens of girls, Lee Anne wanders the club as if it were a party, smiling at men sitting at tables or on sofas.
'Would you like a drink?' asks a middle-aged man with a pleasant demeanour. His name is Dan. Lee Anne sits down and shakes hands, as mannerly as, say, the Queen. They chat about Dallas sports, but time is money, and when the DJ puts on a new record, Lee Anne asks demurely: 'Would this be a good song for me to dance to?'
Dan nods. Lee Anne rises and turns her back so he can unzip her brocade cocktail dress. She strips down to her T-Back (the new-style G-string) while Dan watches in apparent awe. A look of ecstasy on her face, Lee Anne leans over him, caressing her bare breasts.
'I don't feel anything,' she says later, eating a cheeseburger in the dressing room. 'I just take off my clothes and try to become everything he wants me to be.' Only when her parents visited from Wales did she get upset.
'My mum was OK, but my dad wouldn't come to the club. He wanted me to show him what exactly I was doing, so I did . . . at my house and I don't ever want to do it again.' She shudders at the memory, then says: 'Well, they know it's better than being a barmaid in Britain.'
More than 54,000 women dance topless in America, but the girls in the Girl Clubs (they are always 'girls', although in Texas they are sometimes 'ladies') consider themselves an elite. The clubs are ritzy, expensive and have strict rules - barely any touching is allowed between performer and client. Most of the dancers are in their late teens or early twenties; and if in the Eighties they talked about parties and cocaine, in the Nineties, the backstage chat is of healthfood, aerobics and family.
Sky, who works at the Cheetah in Atlanta, is married to 'a man who is in pizzas'. She used to work in computer graphics. Her parents are devout Baptists, and when she told them what she was doing for a living they stopped talking to her. But Taylor, whose mother was also a stripper ('the conditions are a lot better now'), is at ease at Cabaret Royale. 'When I met my husband I was doing this . . . he just kinda fell in love with the whole package. That's who I am and that's how he accepts me.'
With their multi-million-dollar turnover and promotion as respectable venues for hip businessman, the clubs have drawn an inevitable backlash. 'Is this what 20 years of feminism got us?' some women cry. Female executives complain that they are losing business because of deals arranged in clubs that are exclusive or offensive. Sexologists say the clubs are OK, so long as men go with their partners. And middle- class feminist writers such as Susan Faludi suggest 'these new topless clubs come from a desire to make a profit off simmering male anger and fear about women's demands'.
Some men also think the clubs are for macho creeps. They hate the idea of sex as virtual reality - no touching, no intimacy. 'It seemed a vengeful inversion of the old Victorian injunction: ladies don't move,' Keith McWalter wrote in the New York Times. 'Now the men were inert, slouched in their couches and the women were mobile . . . ' For the millions who go to the clubs, however, they are fun. And they are safe.
Maybe the new Girl Clubs belong to the world of the theme park. Old-time burlesque was theatre whose stars - Gypsy Rose Lee, Blaze Starr - worked on a stage. By the Sixties, striptease had become part of pop culture, with strippers flaunting names such as Ineeda Mann and Candy Barr. Then, with the sexual revolution and the arrival of the women's movement, exotic dancing was relegated to seedy backstreet clubs. In the new clubs, it's money that matters. And fantasy - the idea that this girl is doing a table dance just for you.
'The guys are made to feel like gentlemen by the girls who act like ladies,' says the London night-club owner Peter Stringfellow, who is seeking approval to open the first Girl Club in the capital later this year.
'Why do the clubs work?' he muses. 'No rejection. You women gotta go a bit more gentle with us,' he says. 'Our egos are bruised. We can't help it, but we're men, we're programmed to like you people.'
I believe him. But why do we have to dance naked at tables before they can like us?
Well, this is the new age of upmarket sex for sale, when you can order a Sybian Intimate Exerciser ('designed to offer a woman a sensation like straddling a horse saddle, actually a mechanical penis') from a home shopping catalogue for dollars 1,395.
But maybe the last word should go to Justice William Rehnquist. In a case that reached the US Supreme Court, one club protested against a local law concerning nudity. Justice Rehnquist ruled: 'We now hold that the requirement that the dancers in the establishments involved in this case must wear pasties (which cover the nipples) and a G-string, does not violate the First Amendment.'
Oh, yes. When I try to find Lee Anne again, I hear she has left Cabaret Royale to have a baby. Which means she was pregnant when I first met her dancing for Dan.
'Men Only: The Girl Club' will be shown at 10pm next Monday on Channel 4.
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