Anti-CD19 Chimeric Antigen Receptor T-Cell Immunotherapy for Leukemias

Background: * Improved treatments for relapsed and refractory chronic lymphocytic leukemia/Small lymphocytic leukemia (CLL/SLL) and B-cell acute lymphoblastic leukemia or lymphoma (ALL) are needed. * T cells can be genetically modified to express chimeric antigen receptors (CARs) that specifically target malignancy-associated antigens. * Autologous T cells genetically modified to express CARs targeting the B-cell antigen CD19 have caused complete remissions in patients with leukemia or lymphoma. Responses to CAR T-cell therapy in CLL/SLL have historically been lower than in other B-cell malignancies. * CD19 is uniformly expressed on CLL/SLL and ALL. * CD19 is not expressed by normal cells except for B cells, follicular dendritic cells,and some plasma cells. * We have constructed a novel gene therapy construct that encodes a fully-human anti-CD19 CAR. * The conditioning regimen for this trial will include rituximab, fludarabine, and cyclophosphamide. * Possible toxicities include cytokine-associated toxicities such as fever, hypotension, and neurological toxicities. Elimination of normal B cells is probable, and unknown toxicities are also possible. Primary objective, Phase I: -Determine the safety of administering T cells expressing an anti-CD19 CAR with a fully-human single chain variable fragment (scFv) to participants with advanced CLL/SLL(cohort 1) or ALL (cohort 3). Primary objective, Phase II: -Determine the overall response rate (ORR) of a novel conditioning regimen and T cells expressing the Hu19-CD828Z CAR for participants with advanced CLL/SLL and ALL. Eligibility: * Participant must have CD19+ and CD20+ CLL/SLL, or CD19+ and CD20+ B-cell ALL. * Age \>= 18 years of age * Participants must have malignancy that is measurable on a CT scan or by flow cytometry of bone marrow or blood. * Participant must have a creatinine less than 1.5 X institutional ULN and a normal cardiac ejection fraction. Participants with creatinine \>= 1.5 X institutional ULN may participate if serum creatinine eGFR is \>=50 mL/min/1.73m\^2 by 2021 CKD-EPI equation. * An ECOG performance status of 0-1 is required. * No active infections are allowed including hepatitis B or hepatitis C. * Absolute neutrophil count \>= 1000/microL, platelet count \>= 50,000/microL, hemoglobin \>= 8g/dL * Serum ALT and AST less or equal to 3 times the upper limit of the institutional normal unless liver involvement by malignancy is demonstrated. * At least 14 days must elapse between the time of any prior systemic treatment (including corticosteroids) and the first dose of protocol-required rituximab, with the exceptions of Bruton s tyrosine kinase inhibitors (BTKi) for CLL/SLL and tyrosine kinase inhibitors (TKI) for ALL. * Prior CAR T-cell therapy is not allowed. * Demonstration of CD19 and CD20 expression on malignant cells is required for eligibility except if participant has received CD20-directed therapy within 90 days prior to enrollment; in this case, documentation of CD20 expression is not required. * CD19 expression must be "uniform". "Uniform" CD19 expression is defined as no obvious CD19-negative malignant cells being present. CD20 expression, when required for eligibilty, must be demonstrated in at least 20% of malignant cells. Design: * This is a phase I dose-escalation trial with expansion cohorts (Phase II portion) * T cells obtained by leukapheresis will be genetically modified to express the Hu19-CD828Z CAR. * Participants will receive 2 doses of rituximab and a lymphocyte-depleting chemotherapy conditioning regimen with the intent of decreasing the burden of leukemia, which might reduce toxicity and improve anti-leukemia outcomes. * Rituximab will be given in 2 doses, of 375 mg/m\^2 for the first dose and 500 mg/m\^2 (for CLL) or 375 mg/m\^2 (for ALL) for the second dose. * The chemotherapy conditioning regimen is cyclophosphamide 500 mg/m\^2 daily for 3 days and fludarabine 30 mg/m\^2 daily for 3 days. Fludarabine will be given on the same days as the cyclophosphamide. * Three days after the chemotherapy ends, participants will receive an infusion of CAR T cells. * The initial dose level of this dose-escalation trial will be 1.0x10\^6 CAR+ T cells/kg of recipient bodyweight for both the CLL/SLL and the ALL cohorts. * The CAR T-cell cell dose administered will be escalated until a maximum tolerated dose or an optimal dose is determined for each cohort. * Following the T-cell infusion, there is a mandatory 9-day inpatient hospitalization to monitor for toxicity. * Outpatient follow-up is planned for 2 weeks and 1, 2, 3, 4, 6, 9, and 12 months after the CAR T-cell infusion; less frequent follow-up is required more than 1 year after infusion. Long-term gene-therapy follow-up for a total of 15 years after infusion is required.