The objective of this trial is to assess the toxicity of thymus transplantation following unrelated umbilical cord blood transplantation. Emphasis will be placed on adverse events that are not typically associated with umbilical cord blood transplantation. Also, to determine whether engraftment of a third party thymus allograft is feasible in patients who have undergone unrelated umbilical cord blood transplantation. Thymic engraftment will be determined by biopsy of the thymus allograft and immunohistochemical evidence of thymopoiesis. In addition, we would like to assess thymic function by measuring the number of naive T cells and the number of T cell receptor rearrangement excision circles (TRECS), and by assessing the diversity of the T cell receptor beta chain. These data will be compared to age, stem-cell donor source and disease matched historical controls whose stem cell transplants were not followed by a thymus transplant. This will be done in descriptive fashion. Other immune parameters will be followed including T, B, and NK cell numbers, T cell responses to mitogens, antigens, and immunoglobulin production.