Infusion of Allogeneic, 3rd Party CD19-specific T Cells

Study Plan: This study has 2 steps: chemotherapy and gene transfer. The chemotherapy combination in this study (fludarabine and cyclophosphamide) is given to try to help the T-cells work better. Study Groups: If the participant is found to be eligible to take part in this study, the participant will be assigned to a dose level of T-cells based on when the participant joined this study. The first group of 3 participants will receive the lowest dose of T-cells. Each new group will receive a higher dose of T-cells than the group before it, if no intolerable side effects were seen. Up to 6 dose levels of T-cells will be tested. Chemotherapy and Gene Transfer: * On Day -8 (8 days before the T-cell infusion) and Day -7, the participant will receive cyclophosphamide by vein over about 4 hours. the participant will receive mesna by vein over 30 minutes every 4 hours to lower the risk of bleeding in the bladder. * On Day -6 through Day -2, the participant will receive fludarabine 1 time a day by vein over about 30 minutes. Sometime between 2-7 days after chemotherapy ends, the participant will receive the T cell infusion by vein over 15-30 minutes. The infusion will be divided into 2 parts at least 24 hours apart. During and after the infusion, your vital signs will be checked. the participant will be given standard drugs to help decrease the risk of side effects. the participant may ask the study staff for information about how the drugs are given and their risks. If the doctor thinks it is needed based on the status of the disease, the participant may receive up to 2 more courses of this study therapy given at least 6 weeks apart. Study Tests: the participant will be in the hospital from Day -8 through 2 days after the stem cell transplant. Every day while you are in the hospital, blood (about 2 teaspoons) will be drawn for routine tests. About 1 week before the T-cell infusion, the following tests and procedures will be performed. These tests will not be repeated again if the T-cell infusion is repeated. * the participant's medical history will be recorded. * the participant will have a physical exam, including measurement of your vital signs. * Blood (about 4 tablespoons) will be drawn for routine tests. This routine blood draw will include a pregnancy test if the participant is able to become pregnant. * Part of the blood drawn for routine tests will also be used to test for antibodies the participant may develop against the mouse antibodies that are used in the gene transfer process. These antibodies are called human anti-mouse antibodies (HAMA). If your body becomes immune to these proteins, the participant may develop HAMA. The results of this HAMA test will be used to compare against a sample of your blood collected after the transplant is complete, to make sure the participant have not developed an immune system reaction against these mouse protein antibodies. * the participant will have a chest x-ray to check for pneumonia. * the participant will have an ECG to check your heart function. Before the T-cell infusion and up to 2 times within 24 hours after the T-cell infusion, blood (about 4 tablespoons) will be drawn to measure your body's response to the T cell infusion. Follow-Up: Within about 12 hours, 3 days, 1 week, and 2 weeks after the T-cell infusion, the following tests and procedures will be performed. These tests will be repeated again if the T-cell infusion is repeated. * the participant's medical history will be recorded. * the participant will have a physical exam, including measurement of your vital signs. * During the physical exam, you will be checked for possible reactions to the study therapy, such as graft versus host disease (GVHD). GVHD occurs when the donor cells attack the cells of the recipient's body. * the participant will be asked about any side effects you may have had. * Blood (about 4 tablespoons) will be drawn for routine tests and for research to look for the genetically changed T-cells and to count the number of B-cells (other white blood cells) and other non-changed T-cells. Up to 3 times during the first 7 days after the T-cell infusion, blood (about 4 tablespoons) will be drawn to measure your body's response to the T cell infusion. Every 2 weeks during the first month, then 1 time a month until 1 year after the T-cell infusion, blood (2 teaspoons) will be drawn for tests of your immune system. At about 6 weeks, and then 1, 2, and 3 months after the T-cell infusion: * the participant's medical history will be recorded. * the participant will have a physical exam, including measurement of your vital signs. * the participant will be checked for possible reactions to the study therapy, such as GVHD. * the participant will be asked about any side effects you may have had. At about 3 months after the T-cell infusion, blood (about 2 teaspoons) will be drawn to check for HAMA, and you will have breathing tests to check your lung function. At least 1 time during the 3 months after the T-cell infusion: * the participant will have a PET-CT scan and/or a CT scan to check the status of the disease. * If the doctor thinks it is needed, you will have other tests and procedures to check your health, such as a bone marrow aspiration and biopsy. If the disease comes back or a side effect occurs during the 12 months after the T-cell infusion, you may be asked to return for additional follow-up visits if the doctor thinks it is needed. Length of Study: the participant may receive up to 3 courses of study therapy. the participant will no longer be able to receive the study therapy if the disease gets worse, if infections or intolerable side effects occur, or if the participant is unable to follow study directions. the participant may withdraw from the study at any time, but the infusion of the T cells is not necessarily reversible and these types of products have the potential for long-term survival in the body. If the participant withdraw from the study it will only include withdrawal from any procedures specifically preformed for this protocol and long-term follow-up. Please note, long-term follow-up is important for your safety. Your participation on the study will be over once the participant have completed the follow-up visits. Long-Term Follow-Up: For safety reasons, the FDA requires patients receiving gene transfer to have long-term follow-up for at least 15 years after receiving the gene transfer. The participant will be asked to sign a separate consent form (Protocol 2006-0676) for long-term follow up that begins 1 year after this study ends. This is an investigational study. The chemotherapy drugs in this study are commercially available and FDA approved for the treatment of B-cell leukemia or lymphoma. The gene transfer (infusion with genetically modified T-cells) is not commercially available or FDA approved. It is currently being used for research purposes only. Up to 42 patients will take part in this study. All will be enrolled at MD Anderson.