Umbilical cord blood is a source of blood-forming cells that can be used for transplantation. The major problem with this type of transplant is the small number of blood-forming cells available in each cord unit, which may delay the "take" of the graft in the recipient. A strategy to overcome this problem is to give 2 cord blood units, increasing the number of cord blood cells in one of them in the laboratory before they are transplanted. This is done in order to increase their number. The expansion of 1, but not both, of the cord blood units will allow the research team to decide more effectively whether the laboratory expansion is responsible for the speed and content of the "take." At this time, no proof exists that this expansion technique will improve the performance of the cord blood specimens.