This phase II trial studies how well fludarabine and busulfan followed by a donor (allogeneic) stem cell transplant work in treating older patients with acute myeloid leukemia that is in first complete remission. Giving low doses of chemotherapy, such as fludarabine and busulfan, before a donor peripheral blood stem cell transplant helps stop the growth of cells in the bone marrow, including normal blood-forming cells (stem cells) and cancer cells. It may also stops the patient's immune system from rejecting the donor's stem cells. When the healthy stem cells from a donor are infused into the patient they may help the patient's bone marrow make stem cells, red blood cells, white blood cells, and platelets. The donated stem cells may replace the patient's immune system and help destroy any remaining cancer cells. Sometimes the transplanted cells from a donor can make an immune response against the body's normal cells (called graft-versus-host disease). Giving tacrolimus, methotrexate, and rabbit antithymocyte globulin before or after the transplant may stop this from happening.