Anti-CD19 White Blood Cells for Children and Young Adults With B Cell Leukemia or Lymphoma

* INCLUSION CRITERIA * Patient must have a cluster of differentiation 19 (CD19)-expressing B cell ALL or lymphoma and must have relapsed or refractory disease after at least one standard chemotherapy and one salvage regimen. In view of the principal investigator (PI) and the primary oncologist, there must be no available alternative curative therapies and subjects must be either ineligible for allogeneic stem cell transplant (SCT), have refused SCT, or have disease activity that prohibits SCT at this time. * CD19 expression must be detected on greater than 15% of the malignant cells by immunohistochemistry or greater than 30% by flow cytometry in a Clinical Laboratory Improvement Amendments (CLIA) approved test in the Laboratory of Pathology, Center for Cancer Research (CCR), National Cancer Institute (NCI), National Institutes of Health (NIH), or from the referring institution or reference laboratory. The choice of whether to use flow cytometry or immunohistochemistry will be determined by what is the most easily available tissue sample in each patient. In general immunohistochemistry will be used for lymph node biopsies, flow cytometry will be used for peripheral blood and bone marrow samples. * Patients must have measurable or evaluable disease at the time of enrollment, which may include any evidence of disease including minimal residual disease detected by flow cytometry, cytogenetics, or polymerase chain reaction (PCR) analysis. * Greater than or equal to 1 year of age (and at least 15 kg) and less than or equal to 30 years of age. * Adequate absolute CD3 count estimated to be required to obtain target cell dose based on discussion with Department of Transfusion Medicine (DTM) apheresis and Cell Processing Section, DTM. * Subjects with the following central nervous system (CNS) status are eligible only in the absence of neurologic symptoms suggestive of CNS leukemia, such as cranial nerve palsy: * CNS 1, defined as absence of blasts in cerebral spinal fluid (CSF) on cytospin preparation, regardless of the number of WBCs; * CNS 2, defined as presence of \< 5/uL white blood cells (WBCs) in CSF and cytospin positive for blasts, or \> 5/uL WBCs but negative by Steinherz/Bleyer algorithm * CNS3 with marrow disease who has failed salvage systemic and intensive IT chemotherapy (and therefore not eligible for radiation) * Patients with isolated CNS relapse will be eligible if they have previously been treated with cranial radiation (at least centigray (cGy)). * Ability to give informed consent. For subjects \<18 years old their legal guardian must give informed consent. Pediatric subjects will be included in age appropriate discussion and verbal assent will be obtained for those greater than or equal to 12 years of age, when appropriate. * Clinical performance status: Patients \> 10 years of age: Karnofsky greater than or equal to 50%; Patients less than or equal to 10 years of age: Lansky scale greater than or equal to 50%. Subjects who are unable to walk because of paralysis, but who are upright in a wheelchair will be considered ambulatory for the purpose of calculating the performance score. * Patients of child-bearing or child-fathering potential must be willing to practice birth control from the time of enrollment on this study and for four months after receiving the preparative regimen. * Females of child-bearing potential must have a negative pregnancy test because of the potentially dangerous effects on the fetus. * Cardiac function: Left ventricular ejection fraction greater than or equal to 40% by multi-gated acquisition scan (MUGA) or cardiac magnetic resonance imaging (MRI), or fractional shortening greater than or equal to 28% by echocardiogram (ECHO) or left ventricular ejection fraction greater than or equal to 50% by ECHO. * Patients with history of allogeneic stem cell transplantation are eligible if at least 100 days post-transplant, if there is no evidence of active graft versus host disease (GVHD) and no longer taking immunosuppressive agents for at least 30 days prior to enrollment. EXCLUSION CRITERIA Subjects meeting any of the following criteria are not eligible for participation in the study: * Recurrent or refractory ALL limited to isolated testicular disease. * Hepatic function: Inadequate liver function defined as total bilirubin \> 2x upper limit of normal (ULN) (except in the case of subjects with documented Gilbert's disease \> 3x ULN) or transaminase (alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \> 20x ULN based on age and laboratory specific normal ranges; * Renal function: Greater than age-adjusted normal serum creatinine (see below) and a creatinine clearance \< 60 mL/min/1.73 m\^2. * Age: * \<= 5 yrs (Maximum Serum Creatinine = 0.8 mg/dL) * 5 \< age \<=10 yrs (Maximum Serum Creatinine =1.0 mg/dL) * \> 10 yrs (Maximum Serum Creatinine = 1.2 mg/dL) * Hematologic function: * Absolute neutrophil count (ANC) \< 750/microliter, or platelet count \< 50,000/microliter, if these cytopenias are not judged by the investigator to be due to underlying disease (i.e. potentially reversible with anti-neoplastic therapy); * A subject will not be excluded because of pancytopenia greater than or equal to Grade 3 if it is due to disease, based on the results of bone marrow studies. * Hyperleukocytosis (greater than or equal to 50,000 blasts/microliter) or rapidly progressive disease that in the estimation of the investigator and sponsor would compromise ability to complete study therapy; * Pregnant or breast-feeding females; * Recent prior therapy: * Systemic chemotherapy less than or equal to 2 weeks (6 weeks for nitrosoureas) or radiation therapy less than or equal to 3 weeks prior to apheresis; Exceptions: * There is no time restriction in regard to prior intrathecal chemotherapy provided there is complete recovery from any acute toxic effects of such; * Subjects receiving hydroxyurea may be enrolled provided there has been no increase in dose for at least 2 weeks prior to starting apheresis; * Patients who relapse while receiving standard ALL maintenance chemotherapy will not be required to have a waiting period before entry onto this study provided they meet all other eligibility criteria; * Subjects receiving steroid therapy at physiologic replacement doses only are allowed provided there has been no increase in dose for at least 2 weeks prior to starting apheresis; * For radiation therapy: Radiation therapy must have been completed at least 3 weeks prior to enrollment, with the exception that there is no time restriction if the volume of bone marrow treated is less than 10% and also the subject has measurable/evaluable disease outside the radiation port. * Other anti-neoplastic investigational agents currently or within 30 days prior to apheresis (i.e. start of protocol therapy); * Subjects must have recovered from the acute side effects of their prior therapy, such that eligibility criteria are met. Cytopenias deemed to be disease-related and not therapy-related are exempt from this exclusion. * Human immunodeficiency virus/hepatitis B virus/Hepatitis C virus (HIV/HBV/HCV) Infection: * Seropositive for HIV antibody. (Patients with HIV are at increased risk of lethal infections when treated with marrow-suppressive therapy. Appropriate studies will be undertaken in patients receiving combination antiretroviral therapy in the future should study results indicate effectiveness.) * Seropositive for hepatitis C or positive for Hepatitis B surface antigen (HbsAG). * Monoclonal antibody therapy administered within 30 days of the agent prior to apheresis; * Uncontrolled, symptomatic, intercurrent illness including but not limited to infection, congestive heart failure, unstable angina pectoris, cardiac arrhythmia, psychiatric illness, or social situations that would limit compliance with study requirements or in the opinion of the principal investigator (PI) would pose an unacceptable risk to the subject; * Second malignancy other than in situ carcinoma of the cervix, unless the tumor was treated with curative intent at least two years previously and subject is in remission; * History of severe, immediate hypersensitivity reaction attributed to compounds of similar chemical or biologic composition to any agents used in study or in the manufacturing of the cells (i.e. gentamicin);