Antigen Specific Adoptive T Cell Therapy for Adenovirus Infection After Hematopoietic Stem Cell Transplantation

Inclusion Criteria: * Patients must have received allogeneic HSCT and be greater than 30 days post-HSCT at the time of registration. * Patients must have evidence of documented HAdV infection/reactivation. Patients may be: * Symptomatic with any detectable viral load OR * Asymptomatic with viral load that is: \>1000 copies/ml in peripheral blood OR qualitative detection in stool, urine and/or other specimens * Patients must have poor response and/or contraindication to therapy: * Absence of an improvement of viral load (decrease by at least 1 log, i.e. 10-fold) after ≥ 14 days of antiviral therapy with ganciclovir, valganciclovir and/or foscarnet. OR * New, persistent and/or worsening HAdV-related symptoms, signs and/or markers of end organ compromise while on antiviral therapy with ganciclovir, valganciclovir or foscarnet. OR * Have contraindications or experience adverse effects of antiviral therapy with ganciclovir, valganciclovir, cidofovir or foscarnet. * Performance Score: Eastern Cooperative Oncology Group (ECOG) Performance Score ≤ 3. Karnofsky (≥ 16 years) or Lansky (\<16 years) performance score ≥ 50 * The effects of virus-specific, antigen-selected T cells on the developing human fetus are unknown. For this reason, women of child-bearing potential and men must agree to use adequate contraception (double barrier method of birth control or abstinence) 4 weeks prior to study entry, for the duration of study participation and for 3 months after completing treatment. * Subjects who are 14 years and older must have the ability to understand and the willingness to sign a written informed consent document, or assent document. Exclusion Criteria: * Pregnant or breastfeeding women are excluded from this study. Because there is an unknown, but potential risk for adverse events in nursing infants secondary to treatment of the mother with the agents described above, breastfeeding should be discontinued if the mother participates in this trial. * Patients with opportunistic viral infections other than HAdV. * Patients with active, grade II-IV, acute graft versus host disease (GVHD), chronic GVHD or any condition requiring high doses of glucocorticosteroid (\>0.5 mg/kg/day prednisone or its equivalent) as treatment. * Treatment with antithymocyte globulin within 28 days of planned infusion of virus - specific, antigen selected T cells. * Treatment with virus - specific T cells within 6 weeks (42 days) of planned infusion.