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Sharine and Brian Kretchmar of Yukon, Okla., tried a number of medical treatments to conceive a second child. After a depressing series of failures, a doctor finally advised them to find a sperm donor.

 For more than a year, the Kretchmars carefully researched sperm banks and donors. The donor they chose was a family man, a Christian like them, they were told. Most important, he had a clean bill of health. His sperm was stored at the New England Cryogenic Center in Boston, and according to the laboratory’s Web site, all donors there were tested for various genetic conditions.

So the Kretchmars took a deep breath and jumped in. After artificial insemination, Mrs. Kretchmar became pregnant, and in April 2010 she gave birth to a boy they named Jaxon.

But the baby failed to have a bowel movement in the first day or so after birth, a sign to doctors that something was wrong. Eventually Jaxon was rushed to surgery. Doctors returned with terrible news for the Kretchmars: Their baby appeared to have cystic fibrosis.

“We were pretty much devastated,”...