They swim through bright white light, their bodies translucent, tinged with grey and blue. It’s like a crowded plaza down there: some are in a dreadful hurry, others nose about like lazy tourists, a few have stopped altogether. They are the still-warm spermatozoa of a tall, shaven-headed Dane who ejaculated not half an hour ago in a booth at Cryos International Sperm Bank in Arhus, a university town on the northeast coast of Jutland.

Henrietta Peterson leans back from the microscope. Most of the time she studies pigs at the University Hospital of Arhus but two days a week she comes here to count male sperm. Pigs and men, then. “You said it,” she says, carefully. “Not me.”

The count determines the sale price. A sample with five million wriggling sperm per millilitre will sell at €26.4 (£17) per 0.4ml straw. A sample containing 50 million per ml sells for €264 per straw. Sperm is more expensive than gold. Decanted into straws and chilled to minus 196C (-321F), the swimmers will be suspended in a 500litre tank of liquid nitrogen in one of the richest sperm vaults in the world. A year from now, perhaps ten, they will travel by sealed container to a foreign shore where the fastest will revive and swim again. And with this same miraculous, but increasingly common, journey Danish sperm engenders children all over the world.

In the same way that some nations have oil fields or bread mountains, Denmark boasts an ever-growing sperm lake. The vault at Cryos HQ holds around 75,000 straws. It is far too much sperm for a nation where only 65,000 children are born each year, so Denmark is a net exporter. The efforts of the men of Arhus, Odense and Copenhagen have helped to engender an estimated 12,000 children around the world, and each year “the Danish stuff” brings forth some 1,400 more.

An embarrassment of riches in Denmark has corresponded to a scarcity of donor sperm almost everywhere else. In Britain, as in Norway and Sweden, new regulations ending anonymity for sperm donors has decimated the ranks of men once willing to donate, while in April the arrival of the EU Tissue Directive is likely to make sperm banking a harder business to manage on a small scale. Cryos could yet emerge with something of a monopoly on the European market.

The London Bridge centre once supplied donor sperm to most UK fertility clinics. “We now just about meet our own needs,” says Professor Gedis Grudzinskas, medical director. Previously, up to 15 UK clinics relied on semen from Cryos, but such imports are now restricted. “We send our most urgent cases to clinics in Denmark,” says Grudzinskas.

Fertilitetsklinikken Trianglen, north of Copenhagen, is one of these. In the hallway there is a chandelier of sperm-shaped bulbs; on the walls are hundreds of photographs of blonde blue-eyed babies. IVF treatment in this place of candles, rugs and pine floors will cost about £4,000. There are no waiting lists.

West across the Kattegat to Arhus, and Dr Karsten Peterson shows me the Ciconia Private Hospital, another fertility temple of pine floors, rugs and candles. “This place used to be filled with Siemens,” he says. What? “Siemens, the electronic company, had its offices here.” He has already treated 20 British couples. “It’s all very recent, but many are e-mailing me. You can fly Ryanair from Stansted, and we have deals with hotels: fertility tourists get a discount.”

All the doctors I spoke to said that the glut of Danish donors was thanks to a “long tradition of donating” in Denmark, adding that a society where 6 per cent of births involve assisted reproduction technology is likely to be receptive to calls to donate sperm.

Jesper Johansen, 23, president of Copenhagen University Students’ Union, said it was because Danes received a thorough sexual education at primary school. Upon reaching adulthood, as part of national conscription, everyone gets a health check and the option of having their sperm tested. From there it is only a short jump to a donation regime. Then there is Denmark’s liberal attitude to pornography. “Is this your best seller?” I ask a newsagent in Copenhagen, holding up a porn magazine. No one bats an eyelid as the merits of Private and Pige Special are debated — both of which offer “de bedste babes”.

Meanwhile Ole Schou, founder of Cryos, thinks Denmark the perfect location for a sperm bank because “it’s a little country and your neighbour still means something. Eight per cent of Danes give blood,” he says, “and a lot of our sperm donors are also blood donors.” He can take some of the credit for Denmark’s successful donor system. It was while captaining the Danish hang-gliding team that he discovered he also had a talent for organising things. So he went to business school in Arhus, where one night he had a dream of sperm trapped in ice. At the city library he called up everything it had on the subject — he was no scientist, but he found the research papers on sperm compelling. He began examining his own under a microscope, apologising to his family when they found it in the freezer.

Schou brought a new level of efficiency to Danish sperm banking. Only donors with the highest quality sperm — around one applicant in ten — were accepted and Schou also imposed a “daughter test” — “if the interviewing doctor would not want an applicant’s sperm to be used on his daughter, the applicant should be rejected”.

Schou admits this test is highly subjective: it might be the suspicion that an applicant is not telling the truth, it could be just “a sense that this person is not acceptable. Say this guy is not working, he’s sitting, eating crisps in front of the TV all the day, mama and papa doing all the work — I don’t think we would accept him.”