This is a research study of head cooling. Its goal is to determine whether cooling babies' heads can reduce or prevent brain damage that may have resulted from temporarily reduced oxygen supply to the brain. In this study, half of the babies (selected at random) will have a special cooling cap with circulating water placed on their head for 72 hours to lower the temperature of their brain. The rest of the baby's body will be maintained at a defined temperature by a standard overhead radiant heater. The study protocol includes the taking and analysis of blood samples, performance of brain wave tests, imaging of the brain by ultrasound, and other tests as clinically indicated. Neurodevelopmental outcome will also be assessed at 18 months of age.