QUILT-3.006 for Recurrent Medullary Thyroid Cancer

* INCLUSION CRITERIA: Participants must meet the following criteria for participation: * Diagnosis: Patients must have histologically confirmed medullary thyroid cancer by the Laboratory of Pathology or a pathology report and history consistent with medullary thyroid cancer. It is not uncommon for a secondary, minor pathologic focus of another form of thyroid cancer to be coincidentally found in 15-20% of patients with medullary thyroid cancer. In such cases, eligibility is based on the discretion of the investigator. * Patients must have evidence of metastatic medullary thyroid cancer including disease that is evaluable on bone, CT scan or MRI. (Patients who are surgical candidates and potentially rendered disease free with surgical resection are not eligible.) * Patients must have elevated calcitonin levels, greater than 8 pg/mL in females and 16 pg/mL in males * Patients with minimal or no disease related-symptoms (Minimal symptoms will include those that do not affect activities of daily living or pain that does not require regularly scheduled narcotics.) * No brain metastasis, history of seizures, encephalitis, or multiple sclerosis. * Age greater than or equal to 18 years * ECOG performance status of 0-1 at study entry (Karnofsky greater than or equal to 70) * No systemic steroid use within 2 weeks prior to initiation of experimental therapy. Limited doses of systemic steroids to prevent IV contrast, allergic reaction or anaphylaxis (in patients who have known contrast allergies) are allowed. * Hematological eligibility parameters * Granulocyte count greater than or equal to 1,500/mm\^3 * Platelet count greater than or equal to 100,000/mm\^3 * Hemoglobin greater than or equal to 9 g/dL * Biochemical eligibility parameters (within 16 days of starting therapy) * Baseline renal function: \--- Serum creatinine less than or equal to 1.5 x upper limit of normal OR creatinine clearance on a 24-h urine collection of greater than or equal to 60 mL/min. * Hepatic function: * Bilirubin less than or equal to 1.5 mg/dl, in patients with Gilbert's syndrome, a total bilirubin less than or equal to 3.0 mg/dL * AST and ALT\< 2.5 times upper limit of normal * No other active malignancies within the past 3 years (with the exception of nonmelanoma skin cancers, prostate cancer patients with stable biochemical recurrence/not on systemic therapy or carcinoma in situ of the bladder). * Willing to travel to the NIH for follow-up visits * Able to understand and sign informed consent. * Must agree to use effective birth control (such as a condom) or abstinence during and for a period of 6 months after the last vaccination therapy. EXCLUSION CRITERIA: Patients with any of the following will not be eligible for participation in this study: * Patients should have no evidence of immune dysfunction as listed below. * 1 Human immunodeficiency virus (HIV) positivity due to the potential for decreased immune response to the vaccine. * Active autoimmune diseases requiring treatment or a recent history of autoimmune disease requiring therapy, including, but not limited to, inflammatory bowel disease, Crohn's disease, ulcerative colitis, or active diverticulitis. This requirement is due to the potential risks of exacerbating autoimmunity. However, patients with vitiligo may be enrolled. (Patients with history of autoimmune thyroid conditions will be allowed as these patients will be on replacement medications.) * Concurrent use of systemic steroids, except for physiologic doses of systemic steroid replacement or local (topical, nasal, eye drops or inhaled) steroid use. Limited doses of systemic steroids (e.g., in patients with exacerbations of reactive airway disease or to prevent IV contrast allergic reaction or anaphylaxis in patients who have known contrast allergies) are allowed. * Pregnant or breast-feeding women, due to the unknown effects of GI-6207 on the fetus or infant. * Serious inter-current medical illness which would interfere with the ability of the patient to carry out the treatment program. * Untreated brain metastases (or local treatment of brain metastases within the last 6 months) due to the poor prognosis of these patients and difficulty ascertaining the cause of neurologic toxicities. * Patients with pericardial masses \>1 cm or thoracic lesions larger than 2 cm will be excluded. * Concurrent chemotherapy. * Chronic hepatitis infection, including B and C, because potential immune impairment caused by these disorders may diminish the effectiveness of this immunologic therapy. * Participation in another interventional clinical trial at the time of enrollment. * Any significant disease that, in the opinion of the investigator, may impair the patient's tolerance of study treatment. * Significant dementia, altered mental status, or any psychiatric condition that would prohibit the understanding or rendering of informed consent. * Patients with second malignancy within 3 years of enrollment; patients treated surgically with a curative intent, such as non-melanoma skin cancers, localized kidney cancer or carcinoma in situ of the bladder, are not excluded. Patients with MEN2 and a history of pheochromocytoma will also not be excluded. In addition patients with prostate cancer who do not require systemic therapy will not be excluded. (A secondary, minor pathologic focus of another form of thyroid cancer may be coincidentally found in 15-20% of patients with medullary thyroid cancer. In such cases, eligibility is based on the discretion of the investigator.