After the participants receives the donor stem cells, the stem cells divide and reconstitute bone marrow function, blood function, and immunity. The donor stem cells are given after the chemotherapy to shorten the period of low blood counts. They are also given at this time to achieve an antileukemic effect whereby the donor immune cells will recognize the participant's leukemia as "foreign" and prevent its recurrence. A small amount of donor cells will be kept for infusion on a future date (usually 3 and 6 months post transplant) to try to prevent the disease from coming back.