Intraperitoneal Infusion of Autologous Monocytes With Sylatron (Peginterferon Alfa-2b) and Actimmune (Interferon Gamma-1b) in Women With Recurrent or Refractory Ovarian Cancer, Fallopian Tube Cancer or Primary Peritoneal Cancer

* INCLUSION CRITERIA: * Patients must have histologically or cytologically confirmed advanced metastatic or unresectable epithelial ovarian cancer, primary peritoneal cancer or fallopian tube cancer that is relapsed and resistant (recurred less than 6 months after chemotherapy) or refractory (progressed on chemotherapy) to prior platinum- and taxane-based standard care systemic regimen. Or patients who are eligible for aditional platinum therapy. Histopathologic diagnosis must be confirmed in the Laboratory of Pathology (LP), National Cancer Institute (NCI). * Patients must have measurable or evaluable disease. Measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as greater than or equal to 20 mm with conventional techniques or as greater than or equal to 10 mm with spiral computed tomography (CT) scan. * Patients must be at least 4 weeks from previous therapy (chemotherapy, hormonal therapy, and radiation therapy, immunotherapy and monoclonal antibodies, alternative therapy or investigational therapeutic agents). There is no limitation on the amount of prior therapies allowed. Patients with ovarian cancer 4 weeks from previous therapy have been found to have normal monocyte function (unpublished). * Patients who have had cranial radiation therapy need to have completed it greater than or equal to 8 weeks prior to enrollment. * Patients are permitted to receive investigational imaging agents while on study. * Patients who have had major surgery must be fully recovered and require a recovery period of greater than or equal to 4 weeks prior to enrolling on study. * Age greater than or equal to 18 years. Because no dosing or adverse event data are currently available on the use of intraperitoneal monocytes, interferon (IFN)-alpha 2 or IFN-gamma in patients \<18 years of age, children are excluded from this study, but will be eligible for future pediatric trials. * Eastern Cooperative Oncology Group (ECOG) performance status less than or equal to 1 (Karnofsky greater than or equal to 70%). * Adequate renal function, defined as serum creatinine less than or equal to 1.5 X upper limit of normal (ULN), or measured creatinine clearance greater than or equal to 60 mL/min/1.73m\^2. * Adequate hepatic function, defined as aspartate aminotransferase (AST) and alanine aminotransferase (ALT) levels less than or equal to 3 X ULN and total bilirubin \< 1.5 X ULN, unless known diagnosis of Gilbert's syndrome, where bilirubin less than or equal to 5 mg/dl will be permitted. Gilbert's syndrome will be defined as elevated unconjugated bilirubin, with conjugated (direct) bilirubin within the normal range and less than 20% of the total. Total bilirubin will be permitted up to 5 mg/dl, if patients have historical readings consistent with the definition of Gilbert's syndrome prior to entering study. * Adequate bone marrow function, defined as absolute neutrophil (ANC) greater than or equal to 1,500/mm\^3 (greater than or equal to 1.5 X106/L), platelet count greater than or equal to 75,000/mm\^3 (greater than or equal to 75 X10\^6/L), and hemoglobin greater than or equal to 8 g/dL (transfusion to obtain hemoglobin greater than or equal to 8 g/dL is allowed). * The effects of intraperitoneal monocytes, IFN-alpha 2, and IFN-gamma on the developing human fetus are unknown. For this reason and because interferons based on animal data may cause fetal harm, women of child-bearing potential (excludes women with recurrent ovarian cancer) and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately. * Ability of subject to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA: * Patients who are receiving any other investigational agents (with exception of imaging agents as indicated above in the inclusion criteria). * Patients cannot have previously been treated with interferons (e.g., for chronic active hepatitis). * Lack of recovery of prior adverse events to Grade less than or equal to 1 severity (National Cancer Institute Common Terminology Criteria for Adverse Events \[NCI CTCAE\] v 4.03) (except alopecia) due to therapy administered prior to the initiation of study drug dosing. Stable persistent grade 2 peripheral neuropathy may be allowed as determined on a case-bycase basis at the discretion of the Investigator as interferon has not been shown to cause or exacerbate peripheral neuropathy. * Patients with active infection will not be eligible, but may become eligible once infection has resolved and at least 7 days have elapsed after antibiotics use was completed. * Concomitant chronic (daily or almost daily for greater than or equal to 1 month prior) use of steroids or non-steroidal anti-steroidal anti-inflammatory drugs (NSAIDS). * Patients with a recent history (within last 5 years) of autoimmune disease or inflammatory diseases will be excluded, because interferons may worsen these conditions. Exceptions will be allowed for vitiligo and hypothyroidism that has been stable on thyroid replacement medications for \>6 weeks. * Impaired cardiac function or clinically significant cardiac disease including the following: * New York Heart Association class III or IV congestive heart failure * Myocardial infarction within the last 12 months * Subjects known to have impaired left ventricular ejection fraction (LVEF) according to institutional standards * History of allergic reactions attributed to compounds of chemical or biologic composition similar to interferons or other agents used in the study. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations within the last 12 months that would limit compliance with study requirements. Patients with history of neuropsychiatric disorders or Major Depressive Disorder (Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM 5) definition: http://dsm.psychiatryonline.org/doi/full/10.1176/appi.books.9780890425596.dsm04) requiring medical treatment will not be eligible to enroll, based on the black box warning (SYLATRON (peginterferon alfa-2b) for injection, for subcutaneous use. Merck Sharp \& Dohme Corp., a subsidiary of Merck \& Co., Inc., Whitehouse Station, NJ). Exception to this is if patients experienced transient post-partum depression that resolved and patient has been off treatment for \>10 years. Patients who are taking oral anti-depressants for normal sadness, bereavement, or grief will not be excluded. * Pregnant women are excluded from this study because interferons based on animal data may cause fetal harm. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with interferons, breastfeeding should be discontinued if the mother is treated with intraperitoneal interferons. These potential risks may also apply to other agents used in this study. * Patients on combination antiretroviral therapy for the treatment of HIV are ineligible because of the potential for pharmacokinetic interactions with interferons alfa and gamma. * Patients receiving any medications or substances that are potent inhibitors or inducers of Cytochrome P450 1A2 (CYP1A2) or Cytochrome P450 2D6 (CYP2D6) are ineligible.