Brief Summary: This pilot clinical trial studies if cells donated by a close genetic relative can help maintain acute myeloid leukemia (AML) complete remission (CR). Eligible patients will receive a standard induction chemotherapy. If a complete remission results they will receive irradiated allogeneic cells from a HLA haploidentical relative. Only patients who obtain a CR after the standard induction chemotherapy are eligible for the experimental therapy (irradiated haploidentical cells).

Detailed Description: PRIMARY OBJECTIVES: I. Toxicity of haploidentical allogeneic cellular therapy in patients in complete remission (CR) (or CR with incomplete platelet recovery \[CRp\]) after induction chemotherapy with fludarabine (fludarabine phosphate)-cytarabine. II. Efficacy of haploidentical allogeneic cellular therapy in patients in CR (or CRp) after induction chemotherapy with fludarabine-cytarabine (remission rates at 6, 12, 18, 24 months). SECONDARY OBJECTIVES: I. Immunologic parameters before and after haploidentical therapy: host anti-leukemia T cells; host regulatory T cells. OUTLINE: INDUCTION CHEMOTHERAPY: Patients receive fludarabine phosphate intravenously (IV) over 1 hour once daily (QD) for 5 days and cytarabine IV over 4 hours for 5 days. Treatment may continue for 1 or 2 courses at the discretion of the treating physician. ALLOGENEIC CELLULAR THERAPY: Patients undergo irradiated donor lymphocyte infusion (DLI) of 3 x 10\^8 cluster of differentiation (CD)3+ cells/kg at 8 weeks. Patients with stable disease may repeat irradiated DLI every 8-12 weeks in the absence of disease progression or unacceptable toxicity. After completion of study treatment, patients are followed up periodically for up to 2 years.