CDuctive's web site allows the customer to choose from a selection of artists, deciding what tracks, and in what order, they should be laid on the CD.
Once chosen, the customer pays by credit card and the disc is individually pressed and shipped. It takes up to four days to arrive, but works out cheaper than buying from a high street store.
CDuctive is challenging the high price of CDs in Britain, which are more than 50 per cent more expensive than in the US
It hopes to sign up artists who will generate the future big hits. It is already offering a number of tracks by the American rock guitarist Beck - described recently by his peers as one of the most influential musicians now playing - from the days before he signed to his current record label, Geffen.
A CD containing those 17 early Beck tracks, many not otherwise available in the UK, would cost $20.93 (pounds 12.30), including post and packing from the site.
Customers are not obliged to select tracks by a single artist - any combination is possible, with up to 72 minutes of music. The first track costs $4.99, and successive tracks either $0.99 or $0.49, depending on length.
Later this month the company plans to begin advertising at London universities. Students represent its ideal customers; they are likely to have a good Internet connection (free, from the university), want cheap deals, and be interested in upcoming artists who may not yet be signed to a major label.
"The Beck of 18 months from now is probably out there, signed to an independent label," said Alan Manuel, a co-founder of CDuctive, based in New York. "That's who we want to offer on our site."
He said that business is growing at 40 per cent each month since CDuctive started running its site in January, and that about one-third of the orders were international - with more than 80 per cent of those coming from the UK.
"We think it's partly because the language is the same, partly because we have a number of UK artists on the site, and the price is good," he said.
CDuctive is now trying to persuade more major labels to offer their artists' music through the site. The company claims it is a halfway house between traditional mail order - which cuts out the overheads of a shop - and the cutting-edge Internet sites, such as Cerberus which sends songs digitally over the Internet directly to the user.
"The problem there is that you have to buy it over the Internet and collect it over the Internet, and the artists worry about piracy," said Mr Manuel."We think that's too much to deal with."
Instead, "burning" a CD to the customer's demands puts the company, and the artists, in full control, while the buyer only has to get used to passing over their credit details.
"The economics are different for us," said Mr Manuel. "Traditional record labels need to push a lot of units through the shops. They can't promote every artist, so they focus on the ones they reckon will be worth their marketing dollars.
"But we can promote more artists at a lower volume. By making to order, we can afford to show off the artists and promote their music more cheaply."