During the last few decades, research in reproductive technology has been advancing at lightning speeds, and at each juncture along the way—the first “test tube” babies, donor eggs used in post-menopausal women, sperm sorting to choose a baby’s gender, embryo selection to provide donor material for an ill sibling, embryo selection to eliminate the possibility of inherited disease—the question is raised of how much is too much. What is “acceptable” science and medicine? Who decides?

These questions roared back into the headlines this week after the Los Angeles Fertility Institutes announced plans to allow prospective parents to choose the gender, eye, hair,  skin color, and other physical traits of their babies. The service will cost $18,000 per baby and will be available only to couples seeking in-vitro fertilization. The clinic says they have already received five or six requests for the service and expect the first “designer baby” to be born next year.

Trait-selection will be based on a technique called pre-implantation genetic diagnosis, or PGD, created in the 1990s so that parents who carried genes for life-threatening diseases could...