Robots are learning to make human babies. Twenty have already been born.

MEXICO CITY — When she walked into an IVF clinic in June, Alin Quintana knew it would be the last time she would try to conceive a child. She had prepared herself spiritually and mentally for the visit: She had traveled to a nearby state to leave prayer ribbons at a shrine dedicated to Mary Undoer of Knots, a manifestation beloved in Latin America. She had received a blessing from her priest. That morning, as Quintana and her husband wound their way for hours from the unpaved outskirts of the Mexican capital toward the swanky clinic in one of its richest neighborhoods, her mind was alert, as if she were making a recording. She was focusing on the moment, she said, to drown out the weight of painful memories — a lost pregnancy, a burst fallopian tube, emergency surgery, 96 monthly cycles of hope and crushing disappointment.

But now she was ready to be part of a generational experiment: A robot would create her baby. Chatbots have invaded our conversations and self-driving cars roam our streets, but the creation of life is a new frontier for artificial intelligence. In-vitro fertilization has produced more than 13 million babies since...