This is a pilot study designed to evaluate the safety and feasibility of performing umbilical cord blood transplants in older adults or younger infirm patients with high-risk hematopoeitic malignancies. A novel reduced-intensity preparative regimen for umbilical cord blood transplantation will be used. One to a maximum of three cord blood units, depending on cell count, will be administered to facilitate engraftment. Ten patients will be enrolled with an expected accrual rate of 3-4 patients per year and with a goal of completing accrual within 2-3 years.