Trial Evaluating the Efficacy and Safety of Daratumumab in Subjects With Relapsed/Refractory B-cell or T-cell Precursor Acute Lymphoblastic Leukemia (ALL)

Inclusion Criteria: 1. Subject must be at least 18 years of age. 2. The subject must have precursor B-cell or T-cell acute lymphoblastic leukemia. B-cell: relapsed or refractory after first or subsequent salvage therapy; or T-cell: relapsed or refractory with first remission duration less than or equal to 12 months in first salvage; or relapsed or refractory after first or subsequent salvage therapy. 3. More than 5% blasts in bone marrow. 4. Eastern Cooperative Oncology Group (ECOG) performance status \</= 2. 5. Life expectancy of \>/= 12 weeks. 6. Women of childbearing potential must commit to either abstain continuously from heterosexual sexual intercourse or to use 2 methods of reliable birth control simultaneously. This includes one highly effective form of contraception (tubal ligation, intrauterine device, hormonal \[birth control pills, injections, hormonal patches, vaginal rings or implants\] or partner's vasectomy) and one additional effective contraceptive method (male latex or synthetic condom, diaphragm, or cervical cap). Contraception must begin prior to dosing. Reliable contraception is indicated even where there has been a history of infertility, unless due to hysterectomy or bilateral oophorectomy. A man who is sexually active with a woman of childbearing potential must always use a latex or synthetic condom during the study and for 4 months after discontinuing daratumumab. 7. A woman of childbearing potential must have a negative serum or urine pregnancy test at screening within 14 days and again within 72 hours prior to dosing. 8. Each subject must sign an informed consent form (ICF) indicating that he or she understands the purpose of and procedures required for the study and are willing to participate in the study. Subjects must be willing and able to adhere to the prohibitions and restrictions specified in this protocol, as referenced in the ICF. Exclusion Criteria: 1. Active leukemic central nervous system (CNS) disease. 2. Active acute Graft-versus-Host Disease (GvHD) or chronic GVHD grade 2 or higher. 3. Patients who have received prior stem cell transplantation will be allowed to enroll as long as prior transplantation has been at least 3 months before enrollment in the trial and any transplant related toxicities have subsided to Grade 1 or less. 4. Philadelphia chromosome-positive (Ph+) ALL. 5. Cancer chemotherapy within 2 weeks prior to start of daratumumab treatment (steroid or hydroxyurea can be used up to 24 hours prior to first daratumumab infusion for control of high white cell counts) 6. Cancer immunotherapy within four weeks prior to start of daratumumab treatment (exception blinatumomab within two weeks prior) 7. Diagnosed or treated for malignancy other than ALL, except: 1) Malignancy treated with curative intent and with no known active disease present for \>/= 3 years before treatment; 2) Adequately treated non-melanoma skin cancer or lentigo maligna or carcinoma in situ (e.g. cervical, breast) without evidence of disease; 3) or malignancy that in the opinion of the investigator, with concurrence with the MDACC IND office, is considered cured with minimal risk of recurrence within 3 years. 8. Subject has known chronic obstructive pulmonary disease (COPD) with a Forced Expiratory Volume in 1 second (FEV1) \<50% of predicted normal. NOTE: FEV1 testing is required for patients suspected of having COPD and subjects must be excluded if FEV1 \<50% of predicted normal. 9. Subject has known moderate or severe persistent asthma within the past 2 years (see Appendix A: Classification of Asthma Severity), or currently has uncontrolled asthma of any classification. NOTE: subjects who currently have controlled intermittent asthma or controlled mild persistent asthma are allowed in the study. 10. Subject is known to be seropositive for human immunodeficiency virus (HIV), hepatitis B surface antigen, or hepatitis C antibody (unless treated curatively). 11. Subject has any concurrent medical condition or disease (e.g, active systemic infection) that is likely to interfere with study procedures or results, or that in the opinion of the investigator would constitute a hazard for participating in this study. 12. Subject has any of the following laboratory test results at cycle 1 day 1 pre-dosing: 1) Alanine aminotransferase level (ALT) \>/= 2.5 x the upper limit of normal (ULN); 2) Aspartate Aminotransferase (AST) \>/= 2.5 x the ULN; 3) Total bilirubin level \>/= 1.5 x ULN, (except for Gilbert Syndrome: direct bilirubin \>/= 1.5 x ULN); 4) Creatinine \> 2 x ULN. 13. Subject has clinically significant cardiac disease, including: 1) myocardial infarction within 1 year before study enrollment, or an unstable or uncontrolled disease/condition related to or affecting cardiac function (e.g., unstable angina, congestive heart failure, New York Heart Association Class III-IV); 2)uncontrolled cardiac arrhythmia or clinically significant ECG abnormalities; 3) screening 12-lead ECG showing a baseline QT interval as corrected by Fridericia's formula (QTcF) \>470 msec. 14. Subject has known allergies, hypersensitivity, or intolerance to boron or mannitol, corticosteroids, monoclonal antibodies or human proteins, or their excipients (refer to respective package inserts or Investigator's Brochure), or known sensitivity to mammalian-derived products. 15. Subject is a woman who is pregnant, or breast-feeding, or planning to become pregnant while enrolled in this study or within 4 months after the last dose of any component of the treatment regimen. Or, subject is a man who plans to father a child while enrolled in this study or within 4 months after the last dose of any component of the treatment regimen.