Transplantation of Umbilical Cord Blood Following Chemotherapy for Blood Cancers

Inclusion Criteria:

* Patient must have two cord units available. Units must be minimally matched to the subject at 4/6 antigens (HLA Class I (A or B) and Class II (DRB1) - units must have at lease one HLA DRB1 matched allele) and at least one unit must contain a minimum of 1.0 x 107 Total Nucleated Cells /Kg but neither unit may have \> 5 x 107Total Nucleated Cells /Kg. (The feasibility of using particular units will be discussed with the Principal Investigators)
* Disease status precludes waiting to identify a suitably HLA matched unrelated donor
* Patients must have a diagnosis of one of the following:

  * AML
  * refractory AML
  * Secondary AML
  * ALL in CR2 with high-risk features such as short CR1 and/or high-risk cytogenetics
  * ALL in CR1 following initial induction failure
  * Acute mixed lineage leukemia
  * CML beyond chronic phase 1.
  * Lymphoma (Hodgkins or Non-Hodgkins) ineligible for Autologous-BMT
  * Myelodysplastic Syndrome
* Able to provide informed consent or parent/guardian able to provide informed consent.

Exclusion Criteria:

* Consenting 5/6 or 6/6 HLA matched related donor available
* Single cord blood product with cell count \>5 x10E7 Total Nucleated Cells/kg
* Poor Performance Status: ECOG performance status \>= 2 (Karnofsky or Lansky Play performance\<70).
* Poor Cardiac Function (obtained within 3 weeks of the start of transplant): Left ventricular ejection fraction \<= 45% as determined by MUGA or ECHO. For pediatric patients LVEF \< 45 % or a Shortening Fraction below normal limits for age.
* Poor Pulmonary Function (obtained within 3 weeks of the start of transplant):

  * FEV1 and FVC \<50% of predicted for patients who have not received thoracic or mantle irradiation.
  * For patients who have received thoracic or mantle irradiation, FEV1 and FVC \<= 75% of predicted or DLCO \<= 50% of predicted
  * For children unable to perform PFTs second to developmental stage, Pulse oximetry \<= 85% on RA
* Poor Liver Function (obtained within 1 week of the start of transplant): Bilirubin \>= 2.0 mg/dl. (with the exception of patients whose hyperbilirubinemia is the result of Gilbert's disease)
* Poor Renal Function (obtained within 3 weeks of the start of transplant): Corrected CrCl \< 60 mg/min. CrCl will be estimated by the Schwartz formula. A measured CrCl or a GFR may be substituted to determine the subject's CrCl
* HIV Infection: Patients who are HIV positive. (The role of allogenic transplant in HIV+ individuals has not been studied)
* Pregnancy: Patients who are pregnant. (The chemotherapeutic agents used in bone marrow transplant are teratogenic)
* Uncontrolled viral, bacterial or fungal infections
* Patients with symptoms consistent with RSV, influenza A, B or parainfluenza at the time of enrollment on this study will be assayed for the above viruses and if positive are not eligible for the trial until are no longer symptomatic (patients may have continued assay positivity for a period of time post resolution of symptoms second to the nature of the assay)
* Presence of concomitant medication or incident condition that would create an unreasonable risk for the subject to participate in this study as determined by the investigators (Primary or co-investigators).
* Patients with known intolerance to or contraindication for any agent that will be used in the subject's proposed myeloablative or required supportive care regimen.