Prostvac in Patients With Biochemically Recurrent Prostate Cancer

* INCLUSION CRITERIA: * Histopathological documentation of prostate cancer confirmed in either the Laboratory of Pathology at the National Institutes of Health (NIH) Clinical Center, Walter Reed National Military Medical Center, Memorial Sloan Kettering Cancer Center (MSKCC), Dana-Farber Cancer Institute (DFCI) or Beth Israel Deaconess Medical Center (BIDMC) prior to enrollment. If no pathologic specimen is available, patients may enroll with a pathologists report showing a histologic diagnosis of prostate cancer and a clinical course consistent with the disease. * Biochemical progression after definitive radiation or surgery defined as follows: * For patients following definitive therapy: a rise in prostate-specific antigen (PSA) of greater than or equal to 2ng/mL above the nadir (per Radiation Therapy Oncology Group (RTOG)-American Society for Therapeutic Radiology and Oncology (ASTRO) consensus criteria). * For patients following radical prostatectomy: rising PSA after surgical procedure. (Patients must have a PSA greater than or equal to 0.8 ng/ml) * Eastern Cooperative Oncology Group (ECOG) performance status of 0 1 (Karnofsky greater than or equal to 80%). * Patients must have a PSA doubling time of 5-15 months. * Patients must have a rising PSA as confirmed by 3 values done at least 1 week apart and over no less than 1 month. * Recovery from acute toxicity related to prior therapy, including surgery and radiation, or no toxicity greater than or equal to grade 2. * Negative computed tomography (CT) scan/magnetic resonance imaging (MRI) and bone scan for metastatic prostate cancer. * Hematological eligibility parameters (within 16 days before starting therapy) * Granulocyte count greater than or equal to 1000/mm\^3 * Platelet count greater than or equal to 100 000/mm\^3 * Hemoglobin (Hgb) greater than or equal to 10g/dL * Biochemical eligibility parameters (within 16 days before starting therapy): --Hepatic function: bilirubin less than or equal to 1.5mg/dL (OR in patients with Gilbert's syndrome normal. * No other active malignancies within the past 36 months (with the exception of nonmelanoma skin cancers or carcinoma in situ of the bladder) or life-threatening illnesses, in the opinion of the investigator * Willing to travel to the National Institutes of Health (NIH), MSKCC, DFCI, BIDMC for follow-up visits. * 18 years of age or older. * Able to understand and sign informed consent. * Baseline testosterone greater than or equal to 100 ng/dl * PSA less than or equal to 30 ng/mL. * The effects PROSTVAC on the developing human fetus are unknown. For this reason, men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study therapy and at least one month post therapy. Should a woman become pregnant or suspect she is pregnant while her partner is participating in this study, she should inform her treating physician immediately. EXCLUSION CRITERIA: * Immunocompromised status due to: * Human immunodeficiency virus (HIV) positivity. * Active autoimmune diseases such as Addison's disease, Hashimoto's thyroiditis, systemic lupus erythematosus, Sjogren syndrome, scleroderma, myasthenia gravis, Goodpasture syndrome or active Grave's disease. Patients with a history of autoimmunity that has not required systemic immunosuppressive therapy or does not threaten vital organ function including central nervous system (CNS), heart, lungs, kidneys, skin, and gastrointestinal (GI) tract will be allowed. * Other immunodeficiency diseases * Splenectomy * Chronic administration (defined as daily or every other day for continued use \> 14 days) of corticosteroids deemed systemic by investigator within 28 days before the first planned dose of PROSTVAC. Use of inhaled steroids, nasal sprays, intra-articular injections and topical creams for small body areas is allowed. * Serious intercurrent medical illness that, in the judgment of the investigator, would interfere with patient s ability to carry out the treatment program. * Other medications used for urinary symptoms including 5-alpha reductase inhibitors (finasteride and dutasteride) and alternative medications known to alter PSA (eg phytoestrogens and saw palmetto) * History of prior chemotherapy * History of prior immunotherapy within the last 3 years * Major surgery within 4 weeks prior to enrollment (Day 1 visit). * History of allergic reactions attributed to compounds of similar chemical or biologic composition to poxviral vaccines (e.g., vaccinia vaccine) * Known allergy to eggs, egg products, aminoglycoside antibiotics (for example, gentamicin or tobramycin). * History of atopic dermatitis or active skin condition (acute, chronic, exfoliative) that disrupts the epidermis * Previous serious adverse reactions to smallpox vaccination * Unable to avoid close contact or household contact with the following high-risk individuals for three weeks after the Day 1 vaccination: Day 1 vaccination: (a) children less than or equal to 3 years of age, (b) pregnant or nursing women, (c) individuals with current or extensive eczema or other eczemoid skin disorders, or (d) immunocompromised individuals, such as those with HIV. * Receipt of an investigational agent within 28 days (or 56 days for an antibody-based therapy) before the first planned dose of study drugs. * Patients who test positive for hepatitis B virus (HBV) or hepatitis C virus (HCV) * Uncontrolled hypertension (systolic blood pressure (SBP)\>170/ DBP\>105) * Recruitment Strategies This study will be listed on available websites (www.clinicaltrials.gov, https://ccr.cancer.gov/clinical-trials-search-start) and participants will be recruited from the current patient population at NIH.