A Study of Immune Suppression Treatment for People With Sickle Cell Disease or β-Thalassemia Who Are Going to Receive an Allogeneic Hematopoietic Cell Transplantation (HCT)

Inclusion Criteria: * Age ≥ 2 and ≤ 50 years * Suitable haploidentical donor. * Performance score ≥ 70% by Karnofsky Performance Scale or 0 to 1 by ECOG (age \> 16 years), or Lansky Play-Performance Scale ≥ 70% (age ≤ 16 years). * Adequate major organ system function as demonstrated by: * For patients ≥ 18 years of age: * eGFR ≥ 50 mL/min by Cockcroft-Gault formula Formula: ((140 - Age) x Weight (kg)) / (72 x Serum Creatinine (mg/dL) Female Adjustment: Multiply result by 0.85 * For patients \< 18 years of age: * Serum creatinine clearance: glomerular filtration rate \[GFR\]) must be \>50 mL/min/1.73 m2 as calculated by the Schwartz formula * Conjugated (direct) bilirubin less than 3x upper limit of normal. * ALT or AST ≤ 3 times institutional upper limit of normal. * Left ventricular ejection fraction ≥ 50%. * Diffusing capacity for carbon monoxide (DLCO) ≥ 50% predicted, corrected for hemoglobin. For children \< 7 years of age who are unable to perform PFT, oxygen saturation \> 92% on room air by pulse oximetry. * For SCD patients: HbSS, HbSC, HbS/β° with one or more of the following complications: * Acute chest syndrome: 2 or more episodes in the 2 years preceding enrollment * Vaso-occlusive episodes: 3 or more episodes in the 2 years preceding enrollment * Recurrent priapism: 2 or more episodes in the 2 years preceding enrollment * History of osteomyelitis or osteonecrosis * Cerebrovascular disease: * Imaging evidence of prior overt or silent stroke * History of a neurologic event resulting in focal neurologic deficits lasting \> 24 hours * Abnormal transcranial Doppler: Timed average maximum mean velocity ≥ 200 cm/sec in terminal portion of the carotid or proximal portion of the middle cerebral artery or \> 185 cm/sec plus evidence of intracranial vasculopathy if imaging TCD is used * Pulmonary hypertension: Confirmed by right heart catheterization with mean pulmonary arterial pressure ≥ 25 mmHg or mean pulmonary vascular resistance \> 2 Wood units * Red blood cell alloimmunization (\> 3 alloantibodies) * For thalassemia patients: Any genotype, with all of the following: * Onset of red blood cell transfusion dependence during the first 3 years of life * RBC transfusion history \> 225 mL/kg/year or \> 15 lifetime RBC transfusions * Pre-transfusion hemoglobin ≤ 7 g/dL * Hepatosplenomegaly * Patient or the patient's legal representative, parent(s) or guardian should be able to provide written informed consent. Assent of a minor if participant's age is at least seven and less than eighteen years. * For sexually active men and women of childbearing potential, must agree to use a form of contraception considered effective and medically acceptable by the Investigator. Exclusion Criteria: * Prior myeloablative allogeneic HCT. * Overt stroke or CNS instrumentation (e.g. for Moyamoya disease) within 6 months of enrollment. * Liver cirrhosis. Mild fibrosis will be permitted, i.e. fine reticulin or grade 1 of 4, with bridging fibrosis. * Hepatic iron content ≥ 3 mg Fe/g liver dry weight, if applicable * Active hepatitis B or C. * Other uncontrolled infections. * Other malignancy/cancer diagnosis unless in remission after definitive therapy for a minimum of 2 years. Exceptions: Ductal carcinoma in situ, basal cell carcinoma, cervical intraepithelial neoplasia. * Positive pregnancy test in a woman with child-bearing potential, defined as not post-menopausal for 12 months or no previous surgical sterilization. * Inability to comply with medical therapy or follow-up. * Known history of allergic reactions to any constituents of the stem cell product, including a known history of allergic reactions to DMSO.