DISEASE CHARACTERISTICS:

• Diagnosis of 1 of the following hematologic malignancies:

  • Acute myeloid leukemia (AML)

    • In complete remission (CR) by morphology (< 5% blasts within normocellular bone marrow)

    • In second or greater CR OR in high-risk first CR (CR1) as evidenced by any of the following:

      • Preceding myelodysplastic syndromes (MDS)
      • High-risk cytogenetics (e.g., those associated with MDS or complex karyotype)
      • More than 2 courses were required to obtain CR

  • Acute lymphocytic leukemia (ALL) meeting 1 of the following criteria:

    • In high-risk CR1 as evidenced by either of the following:

      • High-risk cytogenetics [t(9;22), t(1;19), t(4;11), or other MLL rearrangements]
      • More than 1 course was required to obtain CR
    • In second or greater CR

  • Chronic myelogenous leukemia (CML)

    • Resistant to imatinib mesylate therapy
    • No active blast crisis

  • Myelodysplasia

    • Blasts < 10% by a representative bone marrow aspirate morphology (otherwise induction chemotherapy required pre-transplantation)

    • Meets 1 of the following criteria:

      • Intermediate-2 or high-risk (e.g., refractory anemia with excess blasts [RAEB] or refractory anemia with excess blasts in transformation [RAEBt]) International Prognostic Scoring System (IPSS) score
      • Refractory anemia with severe pancytopenia
      • High-risk cytogenetics
  • Advanced myelofibrosis

  • Chronic lymphocytic leukemia/small lymphocytic lymphoma (CLL/SLL), marginal zone B-cell lymphoma or follicular lymphoma

    • Progressed after ≥ 2 prior therapies
    • Patients with bulky disease (nodal mass > 5 cm) should be considered for debulking chemotherapy before transplant

  • Lymphoplasmacytic lymphoma, mantle-cell lymphoma, or prolymphocytic leukemia

    • In CR1 or greater or in first partial remission (PR1) or greater after initial therapy

  • Large cell non-Hodgkin lymphoma (NHL)

    • Beyond CR2 or PR2
    • Patients in CR2 or PR2 with initial short remission (< 6 months) are eligible

  • Lymphoblastic lymphoma, Burkitt lymphoma, or other high-grade NHL

    • Patients with stage I or II disease must have progressed within the past year
    • Patients with stage III or IV disease must have completed initial therapy in CR1 or PR1

  • Multiple myeloma

    • Beyond PR2
    • Patients with chromosome 13 abnormalities, first response lasting < 6 months, or β-2 microglobulin > 3 mg/L maybe eligible after initial therapy
• No available HLA-matched sibling donor

• Must have 3 units of partially HLA-matched umbilical cord blood (UCB) available

  • One UCB unit identified as T-regulatory cells source

    • HLA-matched at 4-6/6 HLA antigens with the patient

  • Two UCB units identified as the hematopoietic stem cell (HSC) source

    • Each unit must be HLA-matched at 4-6/6 HLA-A and -B (at low to intermediate resolution) and -DRB1 (at high resolution) antigens with each other
    • The two UCB units must be ABO-matched
    • Total cryopreserved HSC graft cell dose must be > 2.5 x 10^7nucleated cells/kg recipient body weight
• Must also be enrolled on UMN-2000LS068 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:

• Karnofsky performance status 80-100%
• Creatinine ≤ 2.0 mg/dL OR creatinine clearance > 40 mL/min (if creatinine > 1.2 mg/dL or history of renal dysfunction)
• Bilirubin ≤ 2 times normal
• AST and ALT ≤ 2 times normal
• Alkaline phosphatase ≤ 2 times normal
• Pulmonary function > 50% normal
• LVEF ≥ 45%
• No evidence of HIV infection or known HIV-positive serology
• No current active infection
• Not pregnant or nursing

PRIOR CONCURRENT THERAPY:

• See Disease Characteristics
• No steroids prior to UCB infusion