The main purpose of this study is to see if these T-lymphocytes are safe. To make these Ad-specific T lymphocytes the investigators will obtain blood from the stem cell donor and transfer Ad into another type of blood cell, called monocytes. These cells can then stimulate the T lymphocytes and train them to kill cells infected with Ad. The investigators will then grow these Ad-specific T lymphocytes by more stimulation with Ad-infected monocytes and a third type of blood cell called a B lymphoblast from the donor. After testing the T -lymphocytes, the investigators will inject them into patients after transplant who are at high risk of serious Ad virus infection. The investigators will make sure the injected cells are safe and see if they affect the growth and behavior of adenoviruses in the patient's own body.