The purpose of this pilot study is to evaluate the safety and feasibility of infusions of autologous (the patient's own)umbilical cord blood cells in term gestation newborn infants with hypoxic-ischemic encephalopathy. For this study, infants who have signs of moderate to severe encephalopathy at birth whose mothers have previously consented to providing cord blood cells for the Carolinas Cord Blood Bank or other public or private bank that uses accepted standards for collection and handling of cells, or provided verbal consent for cord blood collection for the possibility of their baby's participation in this trial, can receive their own cord blood cells if an adequate number of cells that meet Carolinas Cord Blood Bank Quality standards are available in the first 14 postnatal days. Study activities also include serial blood draws concurrent with clinically indicated blood draws with a total volume of no more than 5 milliliters (1 teaspoon) from all study related tests. Babies will be followed for neurodevelopmental outcome at 4 - 6 and 9 - 12 months at Duke's Special Infant Care Clinic. MRI's will be obtained per clinical routine and results will be analyzed and described in study reports.