Inclusion Criteria

DISEASE CHARACTERISTICS:

- Diagnosis of hematologic malignancy for which a reduced-intensity allogeneic stem
cell transplantation is deemed clinically appropriate, including any of the
following:

- Chronic myelogenous leukemia, meeting one of the following criteria:

- In first chronic phase AND failed imatinib mesylate therapy, defined as
failure to obtain a hematologic remission by 3 months or major cytogenetic
response (Ph+ cells < 35%) by 12 months, or demonstrated clonal evolution
or disease progression while on therapy

- In accelerated phase with < 15% blasts

- In blast crisis that has entered into a second chronic phase following
induction chemotherapy

- Acute myelogenous leukemia, meeting one of the following criteria:

- In second or subsequent completion remission*

- Failed primary induction chemotherapy, but subsequently entered into a
complete remission* with ≤ 2 subsequent re-induction chemotherapy
treatment(s)

- In first complete remission* with poor-risk cytogenetics NOTE: *Complete
remission is defined as < 5% blasts in bone marrow, no definitive evidence
of disease by morphology, flow cytometry, or genetic studies, and no
circulating blasts. Neutrophil and platelet count recovery will not be
required.

- Acute lymphoblastic leukemia, meeting one of the following criteria:

- In second or subsequent complete remission

- In first complete remission AND t(9;22)

- Myelodysplastic syndromes, meeting the following criteria:

- High-risk disease, defined as International Prognostic Scoring System score
of ≥ 1.5

- Less than 10% blasts at the time of study enrollment

- Chronic myelomonocytic leukemia

- Less than 10% blasts at the time of study enrollment

- Myeloid metaplasia with myelofibrosis with poor-risk features, meeting one of
the following criteria:

- Age < 55 years AND a Lille score of 1

- Lille score of 2

- Hemoglobin < 10 g/dL AND abnormal karyotype

- Chronic lymphocytic leukemia/prolymphocytic leukemia, meeting all of the
following criteria:

- Rai stage I-IV disease

- Failed ≥ 1 prior chemotherapy regimen, including fludarabine, or autologous
stem cell transplantation

- Chemosensitive or stable, non-bulky disease prior to transplant

- Received ≤ 3 prior chemotherapy regimens (monoclonal antibody therapy and
involved-field radiotherapy are not considered prior regimens)

- Low-grade B-cell non-Hodgkin lymphoma (NHL) (small lymphocytic lymphoma,
follicular center [grade 1 or 2] lymphoma, or marginal zone lymphoma), meeting
all of the following criteria:

- Failed ≥ 1 prior chemotherapy regimen or autologous stem cell
transplantation

- Chemosensitive or stable, non-bulky disease prior to transplant

- Received ≤ 3 prior chemotherapy regimens (monoclonal antibody therapy and
involved-field radiotherapy are not considered prior regimens)

- Intermediate-grade B-cell or T-cell NHL or mantle cell NHL, meeting all of the
following criteria:

- Failed to achieve remission or recurred after either conventional
chemotherapy or autologous stem cell transplantation

- Chemosensitive, non-bulky disease prior to transplant

- Hodgkin lymphoma, meeting all of the following criteria:

- Relapsed after prior autologous stem cell transplantation or after ≥ 2
combination chemotherapy regimens AND ineligible for autologous peripheral
blood stem cell transplantation

- Chemosensitive, non-bulky disease prior to transplant

- Multiple myeloma, meeting one of the following criteria:

- Relapsed after autologous stem cell transplantation

- Relapsed after conventional therapies AND not a candidate for autologous
stem cell transplantation

- No HLA-matched related or unrelated donor available

- Has two umbilical cord blood units available that are matched at ≥ 4/6 HLA A, B, and
DRB1 with the patient and with each other (HLA C and DQ will not be used in the match
strategy)

- Total combined nucleated cell dose from the 2 umbilical cord blood units must be
> 3.7 x 10^7 nucleated cells/kg (pre-freeze dose) 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%

- Adapted, weighted Charlson Comorbidity Index < 3

- Serum creatinine ≤ 2.0 mg/dL

- AST or ALT < 3 times upper limit of normal (ULN)

- Bilirubin < 1.5 times ULN

- Not pregnant or nursing

- LVEF ≥ 40%

- DLCO > 50%

- No hypoxia at rest with oxygen saturation < 92% on room air (corrected with
bronchodilator therapy)

- No active opportunistic infection (e.g., fungal pneumonia, tuberculosis, or viral
infection)

- No active hepatitis B or C infection that, in the opinion of a gastroenterologist or
the transplant committee, places the patient at moderate- to high-risk for developing
severe hepatic disease

- No HIV infection

PRIOR CONCURRENT THERAPY:

- See Disease Characteristics