Ole Schou, Cryos's director, said that there had been a surge in donations in recent years, allowing the facility to become much more picky about its donors.
"There are too many redheads in relation to demand," he told told Danish newspaper Ekstrabladet. "I do not think you chose a redhead, unless the partner - for example, the sterile male - has red hair, or because the lone woman has a preference for redheads. And that's perhaps not so many, especially in the latter case."
Mr Schou said the only reliable demand for sperm from redheaded donors from Ireland, where he said it sold â€œlike hot cakesâ€.
Cryosâ€™s stores have now reached their peak capacity of 70 litres of semen, and Mr Schou has a waiting list of 600 donors.