Inclusion Criteria

DISEASE CHARACTERISTICS:

- Diagnosis of one of the following:

- Acute myeloid leukemia (AML) meeting 1 of the following criteria:

- Complete first remission (CR1) at high risk for relapse, defined by 1 of
the following:

- Known prior diagnosis of myelodysplastic syndromes (MDS)

- Therapy-related AML

- WBC > 100,000/mm³

- Presence of extramedullary leukemia at diagnosis

- Unfavorable FAB type (M0, M5-7)

- High-risk cytogenetics (those associated with MDS, abnormalities of
Philadelphia chromosomes 5, 7, or 8, or complex karyotype)

- Required ≥ 2 inductions to achieve CR1

- Complete second remission (CR2)

- No AML ≥ CR2

- Acute lymphoblastic leukemia (ALL) meeting 1 of the following criteria:

- CR1 at risk for relapse, defined by 1 of the following:

- WBC > 100,000/mm³ (< 18 years old) OR > 50,000/mm³ (≥ 18 years old)

- Presence of extensive extramedullary disease (excluding CNS disease)

- Presence of high-risk cytogenetic abnormality, such as t(9;22),
t(1;19), t(4;11), or other myelomonocytic leukemia rearrangements
(11q23) or t(8;14) (excluding blastic-phase ALL in pediatric patients)

- Failed to achieve complete remission after 4 weeks of induction
therapy

- CR2 or complete third remission (CR3)

- Not > CR3

- Acute undifferentiated leukemia, infant leukemia, or biphenotypic leukemia in
CR1, CR2, or CR3

- Patients with infant leukemia must be able to receive total-body
irradiation

- Not > CR3

- Juvenile myelomonocytic leukemia with < 30% bone marrow blasts

- Chronic myelogenous leukemia meeting the following criteria:

- Failed prior imatinib mesylate AND in first or second chronic phase or
accelerated phase

- Not in blast crisis

- MDS meeting the following criteria:

- Low (score 0) International Prognostic Scoring System (IPSS) score with
life threatening cytopenia or red cell or platelet-transfusion dependent OR
intermediate (score 1) or high (score 2) IPSS score

- Patients with bone marrow blasts ≥ 10% should have AML induction therapy
with disease response to < 5% blasts and at least partial count recovery

- No MDS with ≥ 10% bone marrow blasts refractory to chemotherapy

- Non-Hodgkin's lymphoma (NHL) meeting 1 of the following criteria:

- High-grade disease in first partial remission (PR) after initial therapy
with biopsy-proven residual disease that is not appropriate for further
chemotherapy or autologous stem cell transplantation

- High-grade or diffuse large cell NHL with recurrent disease after first
remission with chemosensitivity as evidenced by at least PR (defined as >
50% reduction in mass size after therapy)

- No NHL refractory to chemotherapy (less than PR after ≥ 2 regimens)

- No AML evolved from myelofibrosis

- Not a candidate for autologous stem cell transplantation

- No active CNS leukemia

- No bone marrow aplasia (bone marrow cellularity < 5%)

- No acute leukemia with any of the following:

- Morphologic relapse or persistent disease in the bone marrow (cytogenetic
relapse without morphologic evidence of relapse or cytogenetic persistent
disease in the bone marrow is allowed)

- Active extramedullary leukemia

- Required > 2 courses of chemotherapy to obtain present remission status

- No suitably HLA matched related or unrelated volunteer donor available

- Double-unit umbilical cord blood units must meet the following criteria:

- At least 4 of 6 HLA-A and B antigen and DRB1 allele matched with recipient

- At least 3 of 6 HLA-A, B, and DRB1 matched to each other

- Cryopreserved dose of ≥ 1.5 x 10^7 total nucleated cells/recipient body weight
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 (PS) 70-100% (age ≥ 16 years) OR Lansky PS 70-100% (age
< 16 years)

- Creatinine clearance ≥ 60 mL/min OR creatinine < 1.5 mg/dL

- Bilirubin < 2.5 mg/dL (unless Gilbert's syndrome)

- ALT and AST < 3 times upper limit of normal

- Albumin ≥ 2.5 g/dL

- LVEF ≥ 50%

- DLCO ≥ 60%

- No uncontrolled viral, bacterial, or fungal infection

- Not HIV positive

- Not pregnant or nursing

- Negative pregnancy test

- Fertile patients must use effective contraception

PRIOR CONCURRENT THERAPY:

- See Disease Characteristics

- No prior autologous or allogeneic hematopoietic stem cell transplantation

- No prior radiation therapy rendering the patient ineligible for total-body
irradiation