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

Background: \- Although progress has been made in treating children with B-cell cancers such as leukemia or lymphoma, many children do not respond to the standard treatments. One possible treatment involves collecting white blood cells called T cells from the person with cancer and modifying the cells to attack the B-cell cancer. The cells can then be given back to the participant. This study will use T cells that have been modified to attack the cluster of differentiation 19 (CD19) protein, which is found on the surface of some B-cell cancers. Objectives: \- To see if anti-CD19 modified white blood cells are a safe and effective treatment for children and young adults with advanced B-cell cancer. Eligibility: * Children and young adults between 1 and 30 years of age who have B-cell cancer (leukemia or lymphoma) that has not responded to standard treatments. * The leukemia or the lymphoma must have the CD19 protein. * There must be adequate organ function. Design: * Participants will be screened with a physical exam and medical history. Blood and urine samples will be collected. Imaging studies or bone marrow biopsies may be performed depending on the type of cancer. * Participants will undergo a process where white blood cells are collected, called apheresis. These cells will be modified to contain the anti-CD19 gene. * Participants will have 3 days of chemotherapy to prepare their immune system to accept the modified cells. * Participants will receive an infusion of their own modified white blood cells. They will remain in the hospital until they have recovered from the treatment. * Participants will have frequent follow-up visits to monitor the outcome of the treatment. * If the participant benefits from the treatment, then he/she may have the option for another round of treatment.