A BABY GIRL is born in Boston, nearly dead as she enters the world. While still in her mother's womb, she has suffered severe asphyxia, or lack of oxygen. As a result, the baby—called Baby L. in press reports—is born severely underdeveloped. She stays in the hospital for fourteen months. After she goes home, Baby L. must return to the hospital many times to be treated for pneumonia and infections. Her heart stops four times and must be resuscitated, or started again. She frequently requires artificial means—a respirator—to breathe. For the first two years of her life, Baby L. never functions beyond the level of a two- to four-month-old.

Each time Baby L. requires treatment, her mother requests that doctors do everything possible to help her. By the time she is two years old, however, her doctors believe that continuing...