In a hospital room on the Greek island of Crete with views of a sapphire sea lapping at ancient fortress walls, a Bulgarian woman plans to deliver a baby whose biological mother is an anonymous European egg donor, whose father is Italian, and whose birth is being orchestrated from Los Angeles.

She won't be keeping the child. The parents-to-be—an infertile Italian woman and her husband (who provided the sperm)—will take custody of the baby this summer, on the day of birth.

The...