Inclusion Criteria:

- Age 14 to 65 years.

- Available cord blood graft.

- Patients with high risk ALL in first complete remission, with high risk being defined
by the presence of t(4;11), t(9;22) or t(1;19) or patients presenting with extreme
hyperleukocytosis (WBC > 500,000/ul) or partial remission after induction therapy.

- Adult patients with acute non-lymphocytic leukemia (ANLL) in first complete remission
with high-risk cytogenetics (monosomy chromosome 5 or 7, del (5q), abn (3q26),
complex karyotypic abnormalities) or failure to achieve complete remission after
standard induction therapy.

- All patients with ALL or ANLL in second or subsequent remission or partial remission.

- All patients with CML in chronic (failed interferon and/or Gleevec) or accelerated
phase.

- Patients with myelodysplastic syndrome with International Prognostic Scoring System
(IPSS) risk category of INT-1 or greater.

- Patients with severe aplastic anemia must have failed immunosuppressive therapy such
as cyclosporine plus anti-thymocyte globulin.

- Non-Hodgkin's lymphoma or Hodgkin's disease:

- High risk disease in first complete or partial remission

- Chemotherapy-resistant relapse

- Second or subsequent relapse or remission

- Myelofibrosis with myeloid metaplasia.

- High risk, congenital immunodeficiency disorders resulting in recurrent (> 3
episodes) life-threatening infection, known to be curable with allogeneic stem cell
transplantation (to include, but not limited to; severe combined immunodeficiency
disorder, combined immunodeficiency disease, Wiskott-Aldrich syndrome,
Chediak-Higashi syndrome, chronic granulomatous disease, leukocyte adhesion
deficiency, hemophagocytic lymphohistiocytosis).

- Patients with a history of CNS disease must have been treated and have no active CNS
disease at the time of protocol treatment.

- ECOG performance status < or equal to 2.

- Patients must have adequate function of other organ systems as measured by:

- Creatinine clearance (by Cockcroft Gault equation) > or equal to 30 ml/min.
Hepatic transaminases (ALT/AST) < or equal to 4 x normal, bilirubin < or equal
to 2.0 mg/dl

- Pulmonary function tests demonstrating FVC and FEV1 of > or equal to 50% of
predicted for age and DLCO > or equal to 50% of predicted

- Ejection fraction of > or equal to 45% by echocardiogram, radionuclide scan or
cardiac MRI

- Patients must be HIV negative.

- They do not have an HLA-ABC/DR identical related bone marrow or UCB donor.

- They do NOT have a 5/6 antigen matched related bone marrow or UCB donor.

- Their condition precludes waiting to search and find a donor in the National Marrow
Donor Registry or an 8/8 (HLA-A, B, C, DRB1) antigen by high resolution
(allele-level) typing matched unrelated donor was not found.

- Patients must not be pregnant.

Exclusion Criteria:

- Patients that have circulating antibodies specific for donor major histocompatibility
antigens (as determined by panel of reactive antibody assay).

- Patients with progressive ANLL or ALL following second or third-line treatment
regimens.