Inclusion Criteria

DISEASE CHARACTERISTICS:

- Histologically confirmed hematological or lymphatic malignancy, including any of the
following:

- Acute myeloid leukemia

- Relapsed or primary refractory disease with < 10% blasts on peripheral
blood smear

- In first remission with poor risk factors and molecular prognosis [i.e.,
AML with -5, -7, t(6;9), tri8, -11] (preparative regimen 3 or 4)

- Acute lymphocytic leukemia

- In second complete remission or higher OR in first remission with poor risk
factors, including any of the following (preparative regimen 1 or 2):

- BCR/ABL by fluorescence in situ hybridization (FISH) or reverse
transcriptase-polymerase chain reaction

- t(9;22)(q34;q11) detected by cytogenetics

- Chromosomes < 44 by cytogenetics

- DNA index < 0.81 by flow cytometry

- Any rearrangement of chromosome 11 that results in disruption of MLL
gene (11q23) by cytogenetics and SER

- In first remission with poor risk factors and molecular prognosis [ALL with
Philadelphia chromosome-positive t(9;22), t(4;22), (q34;q11)] (preparative
regimen 3 or 4)

- Chronic myelogenous leukemia

- In accelerated phase or greater (preparative regimen 1 or 2)

- In accelerated or second chronic phase (preparative regimen 3 or 4)

- Myelodysplastic syndromes

- With deletion of chromosome 7 or short arm of chromosome 5 (preparative
regimen 1 or 2)

- In high and high-intermediate risk categories (preparative regimen 3 or 4)

- Non-Hodgkin lymphoma in relapse with marrow involvement

- Refractory chronic lymphocytic leukemia

- Patients deemed ineligible for conventional high-dose chemotherapy programs (i.e.,
regimens 1 or 2) due to any of the following concurrent medical conditions may be
eligible for regimens 3 or 4 at the discretion of the treating physician and
principal investigator (preparative regimen 3 or 4):

- LVEF < 50% and > 40%

- FEV1, FVC, or DLCO < 50%

- Bilirubin > 3 mg/dL

- Creatinine > 2 mg/dL

- Two partially HLA-matched umbilical cord blood (UCB) units available

- HLA-matched minimally at 4 of 6 HLA-A, HLA-B, and -DRB1 loci with the patient

- DRB1 matched by high resolution DNA typing

- HLA-A and HLA-B matched by low resolution at the "serological match" level

- Two pooled units with a nucleated cell number > 2.5 x 10^7/kg

- No available HLA-identical sibling or 1 antigen-mismatched related donor

- No available HLA-matched unrelated bone marrow donor

PATIENT CHARACTERISTICS:

- See Disease Characteristics

- Karnofsky performance status (PS) 60-100% OR Lansky PS 60-100% OR Zubrod PS 0-1

- Physiological age 60 or less (at any chronological age)

- Weight > 50 kg

- Creatinine normal for age OR creatinine clearance by 24-hour urine collection or
glomerular filtration rate > 60 mL/min

- Bilirubin ≤ 1.5 mg/dL

- LVEF ≥ 50%

- DLCO ≥ 60% of predicted

- No HIV-1 infection

- No active uncontrolled infection

- Not pregnant

- Negative pregnancy test

- Fertile patients must use effective contraception

PRIOR CONCURRENT THERAPY:

- Recovered from prior intensive chemotherapy