Inclusion Criteria:  Male or female >1 year of age.  Has undergone allogeneic cell transplantation ≥21 days prior to randomization and demonstrated engraftment with an absolute neutrophil count >500/mm^3, AND has one of the following:  AdV viremia DNA ≥10,000 copies/mL at screening, OR  Two consecutive and rising AdV viremia DNA results of ≥1,000 copies/mL at screening, AND  has absolute lymphocyte count <180/mm3, OR has received T cell depletion. Contraceptive use by men and women should be consistent with local regulations regarding the methods of contraception for those participating in clinical studies and refrain from donating sperm or eggs for at least 90 days after treatment completion. Willing and able to provide signed informed consent. Has an HLA type matching with at least 1 suitably matched and available posoleucel VST line for infusion.  Exclusion Criteria:  Grade >2 acute GVHD Ongoing therapy with high-dose systemic corticosteroids Grade 4 diarrhea Uncontrolled viral (other than AdV), bacterial, or fungal infection(s) Requirement for fraction of inspired oxygen >0.5 to maintain arterial oxygen saturation >90% (via pulse oximetry) or need for mechanical ventilation. Prior therapy with anti-thymocyte globulin, alemtuzumab (Campath®), or other immunosuppressive T cell monoclonal antibodies within 28 days prior to randomization. Prior donor lymphocyte infusion or CD34+ stem cell infusion within 21 days prior to randomization. Use of vasopressors within 7 days prior to randomization. Use of any investigational antiviral agent, including brincidofovir, within 7 days prior to randomization. Lactating female unwilling to discontinue nursing prior to randomization. Severe allergy to any component of posoleucel or history of severe prior reactions to blood product transfusions. Positive for SARS-CoV-2 virus at screening.