Low-Dose Fludarabine, Busulfan, and Anti-Thymocyte Globulin Followed By Donor Umbilical Cord Blood Transplant in Treating Patients With Advanced Hematologic Cancer

DISEASE CHARACTERISTICS: * Diagnosis of 1 of the following advanced hematologic malignancies: * Acute myeloid leukemia (AML) meeting the following criteria: * Considered incurable with chemotherapy * Marrow blasts ≤ 10% (may be achieved using standard chemotherapy regimen) * Meets any of the following criteria: * High-risk cytogenetics (-7, -7q, -5, -5q, t(6,9), t(9,11), complex \[≥ 3 abnormalities\], Philadelphia chromosome positive \[Ph+\]) * AML evolved from prior myelodysplasia * AML secondary to prior chemotherapy * Failed to achieve remission * In second or subsequent remission * Refractory relapse * Myelodysplastic syndromes (MDS) meeting the following criteria: * Must have high-risk features, including any of the following: * Intermediate-2 or high risk International Prognostic Scoring System (IPSS) score * Chronic myelomonocytic leukemia * Marrow blasts ≤ 20% (chemotherapy may be given to achieve target blast levels) * No rapidly progressive disease * Acute lymphoblastic leukemia meeting the following criteria: * Considered incurable with chemotherapy * Meets any of the following criteria: * High-risk cytogenetics (Ph+, t(4,11), 11q23 abnormalities, or monosomy 7) * Required \> 1 induction course to achieve remission * Failed to enter remission * In second or subsequent remission * Marrow blasts ≤ 10% (chemotherapy may be given to achieve target blast levels) * Chronic myelogenous leukemia (CML) meeting 1 of the following criteria: * Chronic phase CML that failed imatinib mesylate therapy, as defined by progressive disease or failed to achieve a major cytogenetic response at 1 year after initiation of therapy * Accelerated phase CML meeting 1 of the following criteria: * Failed to achieve a complete cytogenetic remission at 1 year after initiation of therapy * Failed to achieve any cytogenetic response after 6 months of therapy * Progressive disease, as demonstrated by worsening cytogenetic response in 2 consecutive analyses separated by 4 weeks * In blast crisis with \< 10% blasts in bone marrow * Multiple myeloma meeting the following criteria: * Stage I-III disease * Meets any of the following criteria: * In relapse after autologous transplantation * Refractory to ≥ 2 prior conventional myeloma therapies * Chromosome 13 abnormalities (may be enrolled at diagnosis or after initial progression) * Lymphoma * The following subtypes are eligible: * Diffuse large cell * Follicular large cell * Mantle cell * Peripheral T-cell * T-natural killer (T-NK) cell * Hodgkin's lymphoma * Must have progressed, recurred after prior therapy, or failed to respond to primary therapy * Relapsed disease after autologous stem cell transplantation (SCT) allowed * Low-grade non-Hodgkin's lymphoma meeting 1 of the following criteria: * Relapsed or refractory disease after ≥ 2 chemotherapy-based treatment regimens * Relapsed after autologous SCT * Chronic lymphocytic leukemia * Relapsed or refractory disease after ≥ 2 chemotherapy-based treatment regimens * Relapsed after autologous SCT * Meets 1 of the following criteria: * Age 55-70 years * Under age 55 and deemed ineligible for conventional high-dose chemotherapy, as indicated by any of the following: * Poor cardiac function (i.e., LVEF \< 40%) * Poor pulmonary function (i.e., DLCO \< 50%) * Hepatic dysfunction * Prior myeloablative therapy * Not eligible for autologous SCT or conventional therapy * Umbilical cord blood donor available * Matched at ≥ 4 of 6 HLA antigens (A, B, and DR) * Has 1-3 units of umbilical cord blood available * Must not have an HLA-identical or 1 antigen mismatched related donor or potential HLA-matched unrelated donor readily available NOTE: A new classification scheme for adult non-Hodgkin's lymphoma has been adopted by PDQ. The terminology of "indolent" or "aggressive" lymphoma will replace the former terminology of "low", "intermediate", or "high" grade lymphoma. However, this protocol uses the former terminology. PATIENT CHARACTERISTICS: * ECOG performance status 0-2 * Creatinine clearance \> 40 mL/min * Creatinine \< 2.0 mg/dL * AST and alkaline phosphatase \< 3 times upper limit of normal (ULN) * Bilirubin \< 2.0 mg/dL * Hepatitis C or active hepatitis B virus (HBV) allowed if ≤ grade 2 fibrosis and/or inflammation by liver biopsy * Patients with history of HBV infection should be tested for hepatitis B epsilon (HBe) antigen, anti-HBe, and HBV DNA (quantitative) * Patients with active HBV viral replication should receive antiviral therapy * Ejection fraction \> 30% * DLCO ≥ 40% * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception * No active infection requiring ongoing antibiotic treatment * HIV negative PRIOR CONCURRENT THERAPY: * See Disease Characteristics