T Cells Expressing a Novel Fully-Human Anti-BCMA CAR for Treating Multiple Myeloma

* INCLUSION CRITERIA: Multiple Myeloma (MM) criteria: * B Cell Maturation Antigen (BCMA) expression must be detected on malignant plasma cells from either bone marrow or a plasmacytoma by flow cytometry or immunohistochemistry. If patient has plasmacytomas, one plasmacytoma must be biopsied to demonstrate BCMA expression. A specific quantitative level of BCMA expression for eligibility is not specified, but patients with multiple myeloma cells that are negative for BCMA by flow cytometry and immunohistochemistry on either bone marrow biopsy or plasmacytoma biopsy will not be enrolled. These assays must be performed at the National Institutes of Health (NIH). It is not required that the specimen used for BCMA determination comes from a sample that was obtained after the patients most recent treatment. If paraffin embedded unstained samples of bone marrow involved with MM or a plasmacytoma are available, these can be shipped to the NIH for BCMA staining, otherwise new biopsies will need to be performed for determination of BCMA expression. * BCMA expression will need to be documented on the majority of malignant plasma cells by flow cytometry at the NIH at some time after the original anti-BCMA chimeric antigen receptors (CARs) T-cell infusion in all patients undergoing a second anti-BCMA CAR T-cell infusion. * Bone marrow plasma cells must make up less than 50% of total bone marrow cells based on a bone marrow biopsy performed within 24 days of the start of protocol treatment. * Patients must have received at least 3 different prior treatment regimens for multiple myeloma * Must have prior exposure to an immunomodulatory imide drugs ("IMiD") such as lenalidamide and a proteasome inhibitor * Patients must have measurable MM as defined by at least one of the criteria below. * One or more of these abnormalities defines measurable multiple myeloma: * Serum M-protein greater or equal to 1.0 g/dL. * Urine M-protein greater or equal to 200 mg/24 h. * Serum free light chain (FLC) assay: involved FLC level greater or equal to 10 mg/dL (100 mg/L) provided serum FLC ratio is abnormal. * A biopsy-proven plasmacytoma at least 2.0 cm in largest dimension * Bone marrow core biopsy with 30% or more plasma cells Other inclusion criteria: * Greater than or equal to 18 years of age and less than or equal to age 73. * Able to understand and sign the Informed Consent Document. * Clinical performance status of Eastern Cooperative Oncology Group (ECOG) 0-2 * Patients of both genders must be willing to practice birth control from the time of enrollment on this study and for four months after last day of receiving protocol treatment. * Seronegative for human immunodeficiency virus (HIV) antibody. (The experimental treatment being evaluated in this protocol depends on an intact immune system. Patients who are HIV seropositive can have decreased immune-competence and thus are less responsive to the experimental treatment and more susceptible to its toxicities.) * A patient with a negative blood polymerase chain reaction (PCR) test for hepatitis B deoxyribonucleic acid (DNA) test can be enrolled. If hepatitis B DNA (PCR) testing is not available, patients with a negative hepatitis B surface antigen and negative hepatitis B core antibody can be enrolled. * Patients must be tested for the presence of Hepatitis C antigen by PCR and be hepatitis C virus (HCV) ribonucleic acid (RNA) negative in order to be eligible. Only if Hepatitis C PCR testing is not available in a timely manner, patients who are Hepatitis C antibody-negative can be enrolled. * Absolute neutrophil count greater than or equal to 1000/mm(3) without the support of filgrastim or other growth factors within the previous 10 days. * Platelet count greater than or equal to 55,000/mm(3) without transfusion support within the past 10 days. * Hemoglobin greater than 8.0 g/dL * Less than 5% plasma cells in the peripheral blood leukocytes * Serum alanine aminotransferase (ALT) and aspartate aminotransferase (AST) less or equal to 2.5 times the upper limit of the institutional normal. * Serum creatinine less than or equal to 1.5 mg/dL. * Total bilirubin less than or equal to 2.0 mg/dL, except in patients with Gilberts Syndrome who must have a total bilirubin less than 3.0 mg/dL. * At least 14 days must have elapsed since any prior systemic therapy at the time the patient starts the cyclophosphamide and fludarabine conditioning regimen, and patients' toxicities must have recovered to a grade 1 or less (except for toxicities such as alopecia or vitiligo). * Because this protocol requires collection of autologous blood cells by leukapheresis in order to prepare anti-BCMA-CAR T cells, systemic anti-myeloma therapy including systemic corticosteroid steroid therapy of greater than 5 mg/day of prednisone or equivalent dose of another corticosteroid are not allowed within 2 weeks prior to the required leukapheresis. * Normal cardiac ejection fraction (greater than or equal to 50% by echocardiography) and no evidence of hemodynamically significant pericardial effusion as determined by an echocardiogram. * For patients with past participation in gene-therapy, cryopreserved peripheral blood mononuclear cells (PBMCs) that have not been genetically engineered must be available. EXCLUSION CRITERIA: * Patients on any anticoagulants except aspirin. * Patients that require urgent therapy due to tumor mass effects or spinal cord compression. * Patients that have active hemolytic anemia. * Patients with second malignancies in addition to multiple myeloma are not eligible if the second malignancy has required treatment within the past 3 years or is not in complete remission. There are two exceptions to this criterion: successfully treated non-metastatic basal cell or squamous cell skin carcinoma. * Women of child-bearing potential who are pregnant or breastfeeding because of the potentially dangerous effects of the preparative chemotherapy on the fetus or infant. Women of child-bearing potential cannot have a positive pregnancy test. Women of child-bearing potential are defined as all women except women who are post-menopausal or who have had a hysterectomy. Postmenopausal will be defined as women over the age of 55 who have not had a menstrual period in at least 1 year. * Active systemic infections (defined as infections causing fevers or requiring anti- microbial treatment), active coagulation disorders or other major uncontrolled medical illnesses of the cardiovascular, respiratory, endocrine, renal, gastrointestinal, genitourinary, neurologic, or immune system, history of myocardial infarction, active cardiac arrhythmias including active atrial fibrillation history of any arrhythmias other than sinus tachycardia, or atrial fibrillation, currently taking any anti-arrythmic or congestive heart failure medications, active obstructive or restrictive pulmonary disease. * Any form of primary immunodeficiency (such as Severe Combined Immunodeficiency Disease). * Systemic corticosteroid steroid therapy of greater than 5 mg/day of prednisone or equivalent dose of another corticosteroid (prednisone, dexamethasone, etc.) is not allowed within 2 weeks prior to either the required leukapheresis or within 2 weeks prior to CAR T-cell infusion (and at any time after the CAR T cell infusion). * History of severe immediate hypersensitivity reaction to any of the agents used in this study. * Patient unwilling to undergo intensive care unit treatment including mechanical ventilation, cardiopulmonary resuscitation, vasoactive drugs, and hemodialysis. * History of allogeneic stem cell transplantation * Patients with current spinal cord compression (without intradural myeloma involvement). * Patients who have a history (or current evidence) of cerebrospinal fluid multiple myeloma, or intra-dural central nervous system masse * Patients with active autoimmune skin diseases such as psoriasis or other active autoimmune diseases such as rheumatoid arthritis. * Patients must not have required supplemental oxygen within the past month unless it was for a resolved infection. * Patient must not have received genetically modified cells except on prior National Cancer Institute (NCI) gene therapy protocols.