In this study an investigational replication-defective, recombinant adenovirus expressing the interferon-beta gene (BG00001) will be directly injected into tumors, in patients with recurrent Grade III and Grade IV Gliomas, in order to deliver the hIFN-beta gene. The purpose of the study is to evaluate the safety and any harmful effects of injection of BG00001 into brain tumors. Also, this study will help determine whether the virus carrying the beta interferon gene will enter brain tumor cells and cause the cancer cells to die. This study will require one hospital admission for the actual procedure of drug administration. All other visits will be conducted on an out-patient basis