People who have received a solid organ transplant can develop ongoing Epstein-Barr virus (EBV) infection in the blood, which increases the risk of a serious cancer called post-transplant lymphoproliferative disorder (PTLD). This study will test whether rituximab, a drug approved by the U.S. Food and Drug Administration (FDA) for several immune-related conditions, can safely clear EBV from the blood and help prevent PTLD when lowering immune-suppressing medications is not possible or effective. The study includes an initial smaller group focused on determining whether EBV can be cleared, followed by a larger group designed to determine whether treatment lowers the risk of developing PTLD. Researchers will also monitor side effects, transplant organ health, and immune system changes to better understand treatment safety and benefit.