Hypoxic-ischemic encephalopathy (HIE), a condition of reduced blood and oxygen flow to a baby's brain near the time of birth, may cause death or neurologic disability. Cooling therapy (hypothermia) provides some protection, but about half of affected infants still have a poor outcome. This clinical trial will determine if the drug erythropoietin, given with hypothermia, is safe to use as a treatment that may further reduce the risk of neurologic deficits after HIE.