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Karin Cohn carries a genetic defect that led a half-dozen members of her family to develop early breast or ovarian cancer. She firmly supports allowing families such as hers to screen embryos for the defect to ensure no future generations carry it.

But with one daughter born before the genetic test was approved in Britain, she is uneasy with the idea of using the procedure to ensure a second baby in her family is healthy.

"If I didn't have kids now, would I do it? Without question. I think it's an amazing thing," the 38-year-old said. But for her, the prospect of raising a protected child alongside a vulnerable one seems unfair, and she has opted to have no more children.

Both the number of families checking embryos for genetic defects and the number of conditions being tested for are growing rapidly around the world. Last year, the leading U.S. genetic diagnosis clinic—the world's largest—performed more than 1,800 tests aimed at weeding out embryos that carried worrisome family conditions, from sickle cell anemia to cystic fibrosis.

But figuring out the...