Umbilical Cord Blood Stem Cell Transplant in Treating Patients With Hematologic Cancer or Other Disease

DISEASE CHARACTERISTICS:

* Histologically confirmed diagnosis of 1 of the following:

  * Acute myeloid leukemia meeting the following criteria:

    * M0-M7 histologic subtypes by French-American-British classification
    * Previously treated disease
    * Meets 1 of the following criteria:

      * Persistent disease as evidenced by 5-30% persistent blasts in bone marrow after induction or salvage therapy
      * In second or subsequent complete remission (CR)
      * In first CR with 1 of the following high-risk features:

        * Philadelphia chromosome present
        * Noncore-binding factor type of chromosomal abnormalities
  * Myelodysplastic syndromes with 1 of the following International Prognostic Scoring System (IPSS) scores:

    * Intermediate-1
    * Intermediate-2
    * High-risk score with transfusion dependence
  * Chronic myelogenous leukemia meeting 1 of the following criteria:

    * In accelerated or blastic phase
    * Failed prior imatinib mesylate therapy
  * Acute lymphoblastic leukemia meeting 1 of the following criteria:

    * In first CR with any of the following high-risk features:

      * Philadelphia chromosome present
      * Translocation t(4;11) present
      * WBC \> 30,000/mm³ (adult patients)
      * More than 4 weeks from initiation of treatment was required to achieve CR (adult patients)
      * DNA index of near haploid (N=23 chromosomes) (pediatric patients)
    * In second or subsequent CR
    * Persistent disease as evidenced by 5-20% persistent blasts in bone marrow after induction or salvage therapy
  * Hodgkin's or non-Hodgkin's lymphoma meeting the following criteria:

    * Recurrent or refractory disease
    * Tumor ≤ 5 cm in diameter
  * Myeloma or plasma cell neoplasm meeting 1 of the following staging criteria:

    * Stage III at presentation
    * Stage I-II at presentation

      * Not responding OR progressed after first-line therapy
  * Chronic lymphocytic leukemia or Waldenstrom's macroglobulinemia with refractory or progressive disease after first-line therapy
* No 5-6/6 HLA-matched related or 7-8/8 HLA-matched unrelated marrow or peripheral blood stem cell donor available
* No single 4-6/6 HLA-A, -B, or -DRB1-matched umbilical cord blood unit ≥ 3.5 x 10\^7 nucleated cells/kg available

PATIENT CHARACTERISTICS:

* ECOG performance status (PS) 0-2 OR Karnofsky or Lansky PS 70-100%
* Not pregnant
* Fertile patients must use effective contraception prior to and during study participation
* HIV negative
* Bilirubin \< 3.0 mg/dL
* AST and ALT ≤ 3 times upper limit of normal
* Creatinine \< 2.0 mg/dL OR creatinine clearance \> 50 mL/min
* Cardiac ejection fraction \> 50% by echocardiogram OR shortening fraction \> 27%
* No uncontrolled symptomatic congestive heart failure
* No unstable angina pectoris
* No cardiac arrhythmia
* FEV\_1 \> 50% of normal
* Forced vital capacity \> 50% of normal
* DLCO normal
* Oxygen saturation \> 92% on room air (for patients \< 5 years of age)
* No history of allergic reactions attributed to compounds of similar chemical or biological composition to busulfan and fludarabine phosphate
* No ongoing or active infection
* No psychiatric illness or social situation that would preclude study compliance
* No other uncontrolled illness

PRIOR CONCURRENT THERAPY:

* See Disease Characteristics
* At least 4 weeks since prior and no concurrent surgery
* At least 4 weeks since prior and no other concurrent investigational or commercial agents or therapies for the malignancy, including chemotherapy, biologic therapy, or radiotherapy