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.