But at an insurance company called Assicuazion Generali, they were not prepared to give up so easily, and with good reason. The firm was the chief underwriter of a policy taken out by Allen Paulson, Cigar's owner, and Ashford Stud, where he was standing, to cover themselves against just such an eventuality.
The horse's failure to produce the goods in the paddocks cost the insurers $25m (pounds 15m), a record payout for infertility. Now, in the hope of recovering at least some of their losses, they are prepared to try anything to correct the faults in Cigar's reproductive system.
The latest person who will attempt to tackle the problem, the root of which is thought to be a deficiency in Cigar's sperm which makes them misshapen and immobile, is Phil McCarthy, a vet who specialises in stallion reproduction. He believes that Cigar's difficulties may result from the stress of constant competition over the last four years, a period which included a trip to the Middle East to win the inaugural Dubai World Cup, and during which he rarely went for more than a month without a race.
"The best thing we've got going for us now is that we know that performance athletes often have fertility problems after they quit competing," McCarthy says. He has been told that he can do whatever he feels is necessary, with the proviso that the horse must not be harmed in any way. After all, as McCarthy acknowleges, "we must treat him with all the respect that we can because he is an American sporting hero."
His first aim is to relax Cigar as thoroughly as possible, and he has moved his patient to a secluded farm near Paris, Kentucky, where he can live as naturally as possible. The horse who used to put thousands on to the attendance wherever he raced is now seen by very few people, and spends at least 12 hours a day amusing himself in his personal, six-acre paddock.
No one expects quick results, least of all McCarthy, who points out that semen production in horses operates on a 60-day cycle and it could take several cycles before he discovers whether a stress-free existence has had any effect on the shape or mobility of Cigar's sperm.
In the meantime, samples of the horse's semen has been sent to laboratories on three continents for analysis, while experts in the fields of both equine and human fertility have volunteered to assist in the search for a cure for his problem.
McCarthy will be given at least a year, however, to see whether his technique can prove successful. Until then, Cigar will live a life of blissful and absolute relaxation, and since he banked almost $10m (pounds 6m) in prize-money during his brilliant career on the track, no-one could argue that he does not deserve it.