MINNEAPOLIS — The day after Alaina Kilibarda was born, her breathing started to falter, as her family knew it might. During the pregnancy, doctors had told James and Jill Kilibarda that their baby had a lethal genetic problem that would probably end her life within hours of birth.

Most couples choose to have an abortion when they learn that the fetus has a fatal condition. But experts say about 20 to 40 percent of families given such diagnoses opt to carry the pregnancy to term, and an increasing number of them, like the Kilibardas, have turned to programs called perinatal hospice for help with the practical and spiritual questions that arise.

Having learned through the hospice to make the most of the time they had with their child, Alaina’s parents held her and told her things that people reveal to their children spontaneously and haphazardly over a lifetime. Into the October night, as her breathing halted and resumed, they explained how they met in Texas, though both were from Minnesota, and that they fell for each other at first sight. “And we told her that we’ll let her go,” Mrs. Kilibarda said, “and that it’s O.K. to go.”

Traditionally, doctors and nurses dealt with babies born with fatal anomalies by whisking them away from their mothers to die. But in the 1970s, a perinatal bereavement movement began offering parents another way to deal with the death of a child at birth, by acknowledging the grief they feel and by creating family and religious rituals around a stillbirth or early death.