Precursor B Cell Acute Lymphoblastic Leukemia (B-ALL) Treated With Autologous T Cells Genetically Targeted to the B Cell Specific Antigen CD19

Inclusion Criteria: * Adult patients are eligible (\> or = to 18 year old). * Patients must have B- ALL refractory, relapsed, MRD, or in first CR as described below. * Complete remission is defined as restoration of normal hematopoiesis with a neutrophil count \> 1,000 x 106/L, a platelet count \> 100,000 x 106/L, and hemoglobin \> 10 g/dL. Blasts should be \< 5% in a post-treatment bone marrow differential. Furthermore, there should be no clinical evidence of leukemia for a minimum of four weeks. * MRD is defined as patients meeting the criteria for CR above, but with residual disease measured by a quantitative PCR, or by flow or by deep-sequencing of the IgH rearrangements . The assay from blood and/or bone marrow defines MRD by qPCR as a cycle threshold (CT) that is at least 1 CT value \< than the lowest CT value from the background. Outside laboratory tests may suffice for this assessment at the discretion of the Principal Investigator. Relapsed B-ALL will be defined as patients that meet the above criteria for a CR before developing recurrent disease (increased bone marrow blasts). Refractory patients will be defined as patients that have not achieved a CR after 1 cycle of induction chemotherapy * Patients must have a diagnosis of B-ALL by flow cytometry, or bone marrow histology, and/or cytogenetics. * Patients must have CD19+ ALL as confirmed by flow cytometry and/or immunohistochemistry. * Creatinine \< 2.0 mg/100 ml, bilirubin \< 2.0 mg/100 ml, AST and ALT \< 3x normal, PT and PTT \< 2x normal outside the setting of stable chronic anticoagulation therapy. LFTs (Bilirubin, AST, and/or ALT) may be acceptable if the elevation is secondary to leukemia infiltration or leukemia therapy with tyrosine kinase inhibitors. * Adequate cardiac function (LVEF ≥ 40%) as assessed by ECHO or MUGA or other similar cardiac imaging performed within 1 month of enrollment. * Adequate pulmonary function as assessed by ≥ 92% oxygen saturation on room air by pulse oximetry. * Patients must have adequate access for leukapheresis procedure as assessed by staff from the MSKCC Donor Room. * Life expectancy \> 3 months Exclusion Criteria: * Karnofsky performance status \< 70. * Active central nervous system (CNS) leukemia, as defined by unequivocal morphologic evidence of lymphoblasts in the cerebrospinal fluid (CSF) or symptomatic CNS leukemia (i.e. cranial nerve palsies or other significant neurologic dysfunction) within 28 days of enrollment. Prophylactic intrathecal medication is not a reason for exclusion. * Patients previously treated with an allogeneic SCT that is currently complicated by active GVHD requiring T cell suppressive therapy. Patients with following cardiac conditions will be excluded: * New York Heart Association (NYHA) stage III or IV congestive heart failure * Myocardial infarction ≤6 months prior to enrollment * History of clinically significant ventricular arrhythmia or unexplained syncope, not believed to be vasovagal in nature or due to dehydration * History of severe non-ischemic cardiomyopathy with EF ≤20% * Patients with HIV, hepatitis B or hepatitis C infection. * Patients with any concurrent active malignancies as defined by malignancies requiring any therapy other than expectant observation.