Recombinant Measles Virus Vaccine Therapy and Oncolytic Virus Therapy in Treating Patients With Progressive, Recurrent, or Refractory Ovarian Epithelial Cancer or Primary Peritoneal Cancer

Inclusion criteria: * Age ≥ 18 years. * Must have persistent, recurrent or progressive ovarian cancer or primary peritoneal cancer after prior treatment with platinum and taxol compounds. Histologic confirmation of the original primary tumor is required. Prior bilateral oophorectomy is required. * Patients with the following histologic epithelial cell types are eligible: Serous adenocarcinoma, endometroid adenocarcinoma, mucinous adenocarcinoma, undifferentiated carcinoma, clear cell adenocarcinoma, mixed epithelial carcinoma, transitional cell carcinoma, malignant Brenner's Tumor, or adenocarcinoma NOS * The following laboratory values obtained ≤7 days prior to registration: * ANC ≥ 1500/μL * PLT ≥ 100,000/μL * Total bilirubin ≤ upper normal limit * AST ≤ 2 x ULN * Creatinine ≤ 1.5 x ULN * Hgb ≥ 9.0 g/dL * Ability to provide informed consent. * Willingness to return to Mayo Clinic Rochester for follow-up. * Life expectancy ≥ 12 weeks. * Must have anti-measles immunity as demonstrated by serum IgG anti-measles antibody levels of ≥ 20.0 EU/ml as determined by Enzyme Immunoassay (Diamedix, FL). * Must have normal serum CEA levels (\<5 mg/ml) both at the time of study entry and in any prior testing. (NOTE: Not applicable for the MV-NIS cohort.) * Willingness to provide all biologic specimens as required by the protocol. * Measurable disease by exam or CT scan, or, for patients with CA-125 elevation or with microscopic residual but without measurable disease on imaging, willingness to undergo laparoscopy for evaluation of treatment effect if no radiographic progression after 6 treatment cycles. * CD4 count ≥200/μL or ≥15% of peripheral blood lymphocytes Exclusion criteria: * Epithelial tumors of low malignant potential, stromal tumors, and germ cell tumors of the ovary. * Known standard therapy for the patient's disease that is potentially curative or definitely capable of extending life expectancy. Subjects will be excluded if this is their first relapse and they have recurred \>6 mo from completion of primary (adjuvant) chemotherapy. * ECOG performance status (PS) 3 or 4. * Active infection ≤5 days prior to registration. * History of tuberculosis or history of PPD positivity. * History of other malignancy ≤5 years except for non-melanoma skin cancer or carcinoma in situ of the cervix. * Any of the following prior therapies: * Chemotherapy ≤ 3 weeks prior to study entry * Immunotherapy ≤ 4 weeks prior to study entry * Biologic therapy ≤ 4 weeks prior to study entry * Extensive abdominal surgery if it includes enterotomy(ies) \<3 weeks prior to study entry. This criterion does not apply to placement of the peritoneal port-a-cath or lysis of adhesions at the time of study entry. * Any viral or gene therapy prior to study entry * Failure to fully recover from acute, reversible effects of prior chemotherapy regardless of interval since last treatment. * New York Heart Association classification III or IV, known symptomatic coronary artery disease, or symptoms of coronary artery disease on systems review, or known cardiac arrhythmias (atrial fibrillation or SVT). * Requiring blood product support. * CNS metastases or seizure disorder. * HIV-positive test result, or history of other immunodeficiency. * History of organ transplantation. * History of chronic hepatitis B or C. * Other concurrent chemotherapy, immunotherapy, radiotherapy, or any ancillary therapy considered investigational (utilized for a non-FDA-approved indication and in the context of a research investigation). * Any concurrent medications which could interfere with the trial. * Intra-abdominal disease \> 8 cm in diameter at the time of registration, intrahepatic disease, or disease beyond the abdominal cavity. * Treatment with oral/systemic corticosteroids, with the exception of topical or inhaled steroids. * Exposure to household contacts ≤15 months old or household contact with known immunodeficiency. * Allergy to measles vaccine or history of severe reaction to prior measles vaccination. * Allergy to iodine. This does not include reactions to intravenous contrast materials.