Infusion of Allogeneic Umbilical Cord Blood-Derived Cluster of Differentiation Antigen 19 (CD19)-Specific T Cells

Inclusion Criteria:

1. Patients with a history of CD19+ lymphoid malignancies that are primary refractory to treatment (do not achieve complete remission after first course of therapy) or are beyond first remission including second or greater remission or active disease.
2. Patients in first remission are eligible if they are considered high risk, defined as any of the following detected at any time: 1) Acute Lymphoblastic Leukemia (ALL) with translocations 9;22 or 4;11, hypodiploidy, complex karyotype, secondary leukemia developing after cytotoxic drug exposure,and/or evidence of minimal residual disease; or, 2) Acute biphenotypic leukemia; or, 3) Double hit nonHodgkin's lymphoma; or, 4) Non-Hodgkin's Lymphoma (NHL) in second or third complete remission, or relapse (including relapse post autologous hematopoietic stem cell transplant). Double hit lymphomas in first remission or more advanced disease; or, 5) Small Lymphocytic Lymphoma (SLL), or Chronic Lymphocytic Leukemia (CLL) with progressive disease following standard therapy.
3. Age 1 to 75 years old.
4. Performance score of at least 80% by Karnofsky or PS \< 3 (ECOG) (age \>/= 12 years), or Lansky Play-Performance Scale of at least 60% or greater (age \<12 years).
5. Two Cord Blood units identified that are matched with the patient at 4/6, 5/6, or 6/6 HLA class I (serological) and II (molecular) antigens. Each cord must contain at least 1.5 x 10\^7 total nucleated cells/Kg recipient body weight (pre-thaw). One Cord Blood unit may be used (in lieu of two) if it contains at least 2.5 x 10\^7 total nucleated cells/Kg recipient body weight (pre-thaw).
6. Have identified a back up cells source in case of engraftment failure. The source can be autologous, related or unrelated.
7. Cardiac Function: left ventricular ejection fraction \>/= 40%.
8. Pulmonary function: forced expiratory volume at one second (FEV1), forced vital capacity (FVC) and diffusing capacity of lung for carbon monoxide (DLCO) \>/= 50% of expected, corrected for hemoglobin. For children \</= 7 years of age who are unable to perform pulmonary function testing (PFT), oxygen saturation \>/= 92% on room air by pulse oximetry.
9. Renal function: Serum creatinine \</= 1.8mg/dl or \</= 2 x upper limit of normal or creatinine clearance greater or equal than 40 cc/min. Creatinine for pediatric patients \</=1.5 mg/dl or \</=2 times upper limit of normal for age (whichever is less).
10. Liver function: Bilirubin \</= 1.5 mg/dl or \</= 4 x upper limit of normal (unless Gilbert's syndrome), ALT or AST \</= 200 IU/ml or \</= 5 x upper limit of normal for adults unless related to underline disease. For pediatric patients conjugated (direct) bilirubin \< 2x upper limit of normal, ALT or AST \< 5 times upper limit of normal.
11. Negative Beta human chorionic gonadotropin (HCG) test in a woman with child bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization and willing to use an effective contraceptive measure while on study.
12. Patient or patient's legal representative, parent(s) or guardian able to provide written informed consent. Assent of a minor if participant's age is at least seven and less than eighteen years.
13. Patient or patient's legal representative, parent(s) or guardian able to provide written informed consent for the long-term follow-up gene therapy study.

Exclusion Criteria:

1. Positive beta HCG in female of child-bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization or breast-feeding.
2. Patients with known allergy to bovine or murine products.
3. Patients with known history of HIV/AIDS.
4. Patients with chronic active hepatitis or cirrhosis. If positive hepatitis serology, the Study Chair may deem the patient eligible based on the results of liver biopsy.
5. Patients positive for West Nile Virus or RPR.
6. If in the opinion of PI or designee, the research participant has a significant active medical illness or condition deemed to potentially impact negatively on trial participation.