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:

    * Not expected to be curable with chemotherapy and meets ≥ 1 of the following criteria:

      * High-risk cytogenetics (-7, -7q, -5, -5q, t\[6,9\], t\[9,11\], complex, Philadelphia chromosome positive \[Ph+\])
      * AML evolved from prior myelodysplasia
      * AML secondary to prior chemotherapy
      * Failed to achieve remission
      * In second or subsequent remission
    * Marrow blasts ≤ 10% (may be achieved using chemotherapy)
  * Myelodysplastic syndromes (MDS) with high-risk features

    * International Prognostic Scoring System (IPSS) score intermediate -2 or high-risk
    * Marrow blasts ≤ 20% (may be achieved using chemotherapy)
  * Acute lymphoblastic leukemia meeting the following criteria:

    * Not expected to be curable with chemotherapy and meets ≥ 1 of the following criteria:

      * High-risk cytogenetics (Ph+, t\[4,11\], 11q23 abnormalities, and monosomy 7)
      * Required \> 1 induction course to achieve remission
      * Failed to achieve remission
      * In second or subsequent remission
    * Marrow blasts ≤ 10% (may be achieved using chemotherapy)
  * Chronic myelogenous leukemia meeting ≥ 1 of the following criteria:

    * Accelerated phase
    * Chronic phase refractory to imatinib mesylate
    * Blastic phase

      * Marrow blasts ≤ 10% (may be achieved using chemotherapy)
  * Multiple myeloma meeting 1 of the following criteria:

    * Stage II or III disease with \> first relapse or refractory disease
    * Newly diagnosed disease with chromosome 13 abnormalities
  * Lymphoma meeting the following criteria:

    * One of the following subtypes:

      * Diffuse large cell lymphoma
      * Mantle cell lymphoma
      * Peripheral T-cell lymphoma
      * T-natural killer (NK) cell lymphoma
      * Hodgkin's lymphoma
    * Disease failed to respond to primary therapy, progressed, or recurred after prior therapy

      * Patients who have failed autologous stem cell transplantation are eligible provided it has been \> 1 year since transplant
* No rapid progression of malignant disease
* Not eligible for autologous stem cell transplantation
* Available umbilical cord blood (1-3 units) donor matching at ≥ 4 of 6 HLA antigens (A, B, and DR)

  * Patients with an HLA-identical or 1 antigen-mismatched related donor OR a potential HLA-matched unrelated donor matching at \> 6/8 (A, B, C, DR) alleles are not eligible

PATIENT CHARACTERISTICS:

* ECOG performance status 0-2
* Creatinine \< 2.0 mg/dL
* Creatinine clearance \> 40 mL/min
* Bilirubin \< 2.0 mg/dL
* AST and alkaline phosphatase \< 3 times upper limit of normal
* Hepatitis C and active hepatitis B allowed if patient has ≤ grade 2 inflammation or fibrosis by liver biopsy
* Ejection fraction \> 40% by echocardiogram or MUGA
* DLCO \> 40% of predicted
* Not pregnant or nursing
* Negative pregnancy test
* No known HIV infection
* No active infection requiring ongoing antibiotic treatment

PRIOR CONCURRENT THERAPY:

* See Disease Characteristics