Intravenous and Intraperitoneal Paclitaxel, Intraperitoneal Cisplatin, and Intravenous Bevacizumab for the Initial Treatment of Optimal Stage II or III Ovarian, Primary Peritoneal, and Fallopian Tube Cancer

Inclusion Criteria: * Subjects must have signed an approved informed consent. * Subjects with histologic diagnosis of epithelial ovarian carcinoma, primary peritoneal carcinoma, or fallopian tube carcinoma, Stage II or III, with optimal (≤ or equal to 1 cm residual disease) residual disease following initial surgery. All subjects must have had appropriate surgery for ovarian, primary peritoneal, or fallopian tube carcinoma with appropriate tissue available for histologic evaluation to confirm diagnosis and stage. Pathology must be verified at Memorial Sloan-Kettering Cancer Center. * Subjects with the following histologic epithelial cell types are eligible: Serous adenocarcinoma, endometrioid adenocarcinoma, mucinous adenocarcinoma, undifferentiated carcinoma, clear cell adenocarcinoma, mixed epithelial carcinoma, transitional cell carcinoma, malignant Brenner's Tumor, or adenocarcinoma N.O.S. * Subjects must have a Karnofsky Performance Status (KPS) of ≥ or equal to 70%. * Subjects must be entered no more than 12 weeks postoperatively. * Bone marrow function: * Absolute neutrophil count (ANC) ≥ than or equal to 1,500/µl (equivalent to Common Toxicity Criteria (CTC) Grade 1) * Platelets ≥ than or equal to 100,000/µl (CTC Grade 0-1) * Renal function: Creatinine ≤ than or equal to 1.5 mg/dl * Hepatic function: Bilirubin ≤ than or equal to 1.5 x ULN (CTC Grade 1) AST ≤ than or equal to 2.5 x ULN (CTC Grade 1) * Neurologic function:Neuropathy (sensory) ≤ than CTC Grade 1 * Urine Protein Creatinine: Urine protein creatinine (UPC) ratio must be \< than 1.0 gm. If UPC ratio \> than or equal to 1, collection of 24-hour urine measurement of urine protein is recommended as part of the patient's medical management off-study. \* * UPC ratio of spot urine is an estimation of the 24 urine protein excretion - a UPC ratio of 1 is roughly equivalent to a 24-hour urine protein of 1 gm. UPC ratio is calculated using one of the following formulas: * \[urine protein\]/\[urine creatinine\] - if both protein and creatinine are reported in mg/dL * \[(urine protein) x0.088\]/\[urine creatinine\] - if urine creatinine is reported in mmol/L * The UPCR has been found to correlate directly with the amount of protein excreted in a 24 hour urine collection. Specifically, a UPCR of 1.0 is equivalent to 1.0 gram of protein in a 24 hour urine collection. Obtain at least 4 ml of a random urine sample in a sterile container (does not have to be a 24 hour urine). Send sample to lab with request for urine protein and creatinine levels \[separate requests\]. The lab will measure protein concentration (mg/dL) and creatinine concentration (mg/dL). The UPCR is derived as follows: protein concentration (mg/dL)/creatinine (mg/dL). Patients must have a UPCR \< 1.0 to allow participation in the study. * Blood coagulation parameters: PT such that international normalized ratio (INR) is \< than or equal to 1.5 (or an in-range INR, usually between 2 and 3, if a patient is on a stable dose of therapeutic warfarin) and a PTT \< 1.2 times the upper limit of normal. * Patients of childbearing potential must have a negative serum pregnancy test prior to study entry and be practicing an effective form of contraception during the study and for at least 6 months after receiving the final treatment of bevacizumab. * Patients must have an Intraperitoneal (IP) port in place. If a patient does not have an IP port, she must be willing to undergo surgical placement of one. Exclusion Criteria: * Subjects with a current diagnosis of epithelial ovarian tumor of low malignant potential (borderline carcinomas) are not eligible. Subjects with a prior diagnosis of a low malignant potential tumor that was surgically resected and who subsequently develop invasive adenocarcinoma are eligible, provided that they have not received prior chemotherapy for any ovarian tumor. * Subjects who have received prior radiotherapy to any portion of the abdominal cavity or pelvis are excluded. Prior radiation for localized cancer of the breast, head and neck, or skin is permitted, provided that it was completed more than 3 years prior to enrollment, and the subject remains free of recurrent or metastatic disease. * Subjects who have received prior chemotherapy for any abdominal or pelvic tumor are excluded. Subjects may have received prior adjuvant chemotherapy for localized breast cancer, provided that it was completed more than 3 years prior to enrollment, and that the subject remains free of recurrent or metastatic disease. * Patients with synchronous primary endometrial cancer, or a history of primary endometrial cancer, are excluded unless all of the following conditions are met: 1. Stage not greater than IB. 2. Less than 3 mm invasion without vascular or lymphatic invasion 3. No poorly differentiated subtypes, including papillary serous, clear cell or other FIGO grade 3 lesions * Patients with suboptimal (\> 1 cm) residual disease, as determined by the operative surgeon. * Patients who have received any targeted therapy (including but not limited to vaccines, antibodies, tyrosine kinase inhibitors) or hormonal therapy for management of their ovarian, peritoneal primary, or fallopian tube cancer. * With the exception of non-melanoma skin cancer and other specific malignancies as noted above, subjects with other invasive malignancies who had (or have) any evidence of the other cancer present within the last 5 years or whose previous cancer treatment contraindicates this protocol therapy are excluded. * Subjects with acute hepatitis. * Subjects with active infection that requires parenteral antibiotics. * Patients with serious, non-healing wound, ulcer, or bone fracture are not eligible. This includes history of abdominal fistula, gastrointestinal perforation or intra-abdominal abscess within 28 days. Patients with granulating incisions healing by secondary intention with no evidence of fascial dehiscence or infection are eligible but require weekly wound examinations. * Patients with active bleeding or pathologic conditions that carry high risk of bleeding, such as known bleeding disorder, coagulopathy, or tumor involving major vessels. * Patients with history or evidence upon physical examination of CNS disease, including primary brain tumor, seizures not controlled with standard medical therapy, any brain metastases, or history of cerebrovascular accident (CVA, stroke), transient ischemic attack (TIA) or subarachnoid hemorrhage within 6 months of the first date of treatment on this study. * Patients with clinically significant cardiovascular disease. This includes: 1. Uncontrolled hypertension, defined as systolic \>150 mm Hg or diastolic \> 90 mm Hg 2. Myocardial infarction or unstable angina \< 6 months prior to registration 3. New York Heart Association (NYHA) Grade II or greater congestive heart failure 4. Serious cardiac arrhythmia requiring medication 5. CTCAE grade 2 or greater peripheral vascular disease * Patients with major surgical procedure, open biopsy, laparoscopy (including intraperitoneal port placement) or significant traumatic injury within 28 days prior to the first date of bevacizumab therapy. Major surgical procedure anticipated during the course of the study. Minor surgical procedures, fine needle aspirates, or core biopsies within 7 days prior to the first date of bevacizumab therapy. * Patients with known hypersensitivity to Chinese hamster ovary cell products or other recombinant human or humanized antibodies. * Patients under the age of 18. * Patients who are pregnant or nursing. * Evidence of extensive intraperitoneal adhesions at the time of surgery, as determined by the operative surgeon.