OUTLINE: This is a randomized study. Patients are stratified according to gross residual disease (present vs absent) and whether second-look surgery will be performed at the end of treatment (yes vs no). Blood is drawn for BRCA mutation analysis and DNA extraction before the start of chemotherapy, but after randomization. Patients are randomized to one of two treatment arms. Patients in arm I receive IV paclitaxel by 24-hour infusion on day 1 followed by IV cisplatin on day 2. Patients in arm II receive IV paclitaxel by 24-hour infusion on day 1 followed by intraperitoneal (IP) cisplatin on day 2, plus IP paclitaxel on day 8. Treatment for both arms repeats every 3 weeks for a total of 6 treatment courses. Following chemotherapy, second look surgery is performed if selected by the patient. Quality-of-life assessments are performed prior to randomization, prior to course 4, 3-6 weeks after the completion of course 6 and prior to second look surgery if selected, 6 months after treatment is completed, and 12 months after treatment is completed. Patients are followed every 3 months for 2 years, then every 6 months thereafter.