The purpose of this study is to test the safety of a transfusion of specialized white cells from your transplant donor's blood, called T-cells, that have been grown and immunized against the CMV virus in the test tube. If the transplant donor is immune to CMV (ie: the donor has antibody to CMV in the blood), the T-cells will be selected and grown from the blood of the transplant donor. However, if the transplant donor is not immune to CMV, or if T-cells from the donor are not readily available, CMV-immune T-cells grown from the blood of another normal donor who is partially matched to the patients tissue type can be used. The transplant physician will explain which of these treatments is available to the patient. This trial is called a phase I trial because phase I trials are designed to test the safety of different doses of an experimental treatment. We want to find out what effects, good and/or bad, a dose/doses of these immune T-cells will have on the patient and on the CMV infection. Specifically, we wish to test CMV immune T-cells grown from your blood using a new method developed at our center. In this method, fragments of an important CMV protein, called CMVpp65, are chemically synthesized and then used to immunize T-cells in the test tube.