They thought their embryo didn't take. Then their son was born to a stranger across the country, lawsuit claims.

About a decade ago, Anni and Ashot Manukyan of Glendale, Calif., entrusted their embryos to a fertility clinic in Los Angeles that promised its in vitro fertilization technique would lead to a baby.

When they returned last year to try for a pregnancy, one of the embryos took. But they say it was placed in the uterus of the wrong woman, who then gave birth to their son across the country in New York.

When the couple discovered the error this spring, they said it took an agonizing lega...