Chicago White SoxAlexei Ramirez’s gold medal replica.

The IOC keeps medal molds from modern Games in the Olympic Museum in Lausanne, Switzerland, a spokeswoman says. She adds the organization, which has 34,237 medalists in its database, gets one or two replacement requests every year. The replacements have the word replica on them, usually in tiny print on the bottom edge.

The U.S. Olympic Committee says replicas generally cost the athlete between $500 and $1,200, depending on the intricacy of the design.

Getting an Olympic replica takes months. Alexei Ramirez, a Chicago White Sox shortstop who won gold for Cuba’s baseball team in 2004, says someone stole his medal as he and his wife relocated to the U.S. The White Sox sent the IOC a police report and payment this past spring. Two months later, the team received the new medal—without a strap, since the IOC doesn’t supply replica ribbons—via DHL and surprised Mr. Ramirez with an on-field presentation.

Mr. Ramirez says he keeps his replica in a safe place, but he won’t say exactly where. “That’s a secret,” he says. “I’m not going to tell anybody to make sure it doesn’t get stolen again.”

Some Olympians don’t like talking about their absent-minded moment. Glenn Eller, a shotgun shooter who won gold in Beijing, says only that someone took it while he was out with colleagues in Fort Worth, Texas, in late 2008. “I put myself in a situation that I probably shouldn’t have been in, and someone stole it out of my pocket,” he says. “I’m trying to forget it and go ahead.” He has since received a replica.

Olympic officials warn it can be tough to replicate certain medals if they contain materials other than metal. U.S. water polo goalie Merrill Moses, who had his silver from the 2008 Beijing Games stolen in a burglary of his parents’ house, says his replica medal contained jade that looked painted on, rather than a piece embedded in the back.

Mr. Moses returned that replica to Olympic officials, who told him they found a way to make a better one. In the meantime, he is toting around something else: a $75 knockoff silver medal he bought on eBay. “I do a lot of camps and clinics…and the kids want to see a medal,” Mr. Moses says, adding that he tells them it isn’t the real thing.

Before there was an official process for getting replacement medals, athletes made do with makeshift ones. Olympics historian David Wallechinsky says Canadian high jumper Duncan McNaughton lost his 1932 gold medal. So his friend Bob Van Osdel—the high-jump runner-up who happened to be a dentist—made a mold from his silver medal, filled it with gold and sent the replica to Mr. McNaughton, the historian says.

Corey Codgell, a shotgun shooter who won bronze in Beijing, doesn’t take any chances. She usually keeps her nicked-up medal in her front pocket when she travels. Before letting an audience at an event handle it, she warns everybody: “No one leaves this room until I get my medal back.”