A Study of Recombinant Vaccinia Virus to Treat Unresectable Primary Hepatocellular Carcinoma

Inclusion Criteria * Histological confirmation or clinical/laboratory diagnosis of primary hepatocellular carcinoma (HCC) * Cancer is not surgically resectable for cure * Child Pugh A or B * Tumor progression during or after at least one prior HCC treatment regimen (Note: If standard HCC therapies are either medically contraindicated or patient has refused those treatments, the patient may be eligible for this study) * Performance Score: KPS score of ≥ 70 * Anticipated survival of at least 16 weeks * Total bilirubin ≤ 2.5 x ULN * AST, ALT \< 5.0 x ULN * WBC \> 2,500 cells/mm3 and \< 50,000 cells/mm3 (GCSF treatment allowed) * ANC \> 1,250 cells/mm3 (GCSF treatment allowed) * Hemoglobin ≥ 9 g/dL (RBC transfusion allowed) * Platelet count ≥ 50,000 plts/mm3 * Acceptable coagulation status: INR ≤ 1.5 x ULN * Acceptable kidney function: Serum creatinine \< 2.0 mg/dL * If patients are diabetic or have a screening random glucose \> 160 mg/dL, a fasting glucose must be done and patients must be WNL or Grade 1 in order to be eligible for the study * For patients who are sexually active: able and willing to abstain from sex during treatment period and for 3 weeks following treatment, and use an acceptable method of birth control for 3 months after last injection with JX-594 * Able/willing to sign an IRB/IEC/REB-approved written consent form * Able and willing to comply with study procedures and follow-up examinations, including compliance with the "Infection Control Guidelines for Patients" (in written consent form) Exclusion Criteria: * Current, known extra-hepatic tumors that, in the investigator's medical opinion, are likely to result in significant morbidity or mortality within the next 16 weeks. * Pregnant or nursing an infant * Known infection with HIV * Clinically significant active infection or uncontrolled medical condition considered high risk for investigational new drug treatment * Significant immunodeficiency due to underlying illness (e.g. hematological malignancies, congenital immunodeficiencies and/or HIV infection/AIDS) and/or medication (e.g. high-dose systemic corticosteroids) * History of exfoliative skin condition (e.g. severe eczema, ectopic dermatitis, or similar skin disorder) that at some stage has required systemic therapy * Clinically significant and/or rapidly accumulating ascites, peri-cardial and/or pleural effusions * Liver tumors in a location that would potentially result in significant clinical adverse effects if post-treatment tumor swelling were to occur (e.g. tumors impinging on the biliary tract that could affect drainage) * Severe or unstable cardiac disease * Current, known CNS malignancy * Anti-cancer therapy (e.g. RFA, TACE, PEIT, radioembolization, chemotherapy, surgery, or an investigational drug, etc.) within 4 weeks prior to first treatment * Absolute contraindication to undergoing MRI scanning * Experienced a severe systemic reaction or side-effect as a result of a previous smallpox vaccination * Use of anti-platelet or anti-coagulation medication * Use of the following anti-viral agents: ribavirin, adefovir, cidofovir (within 7 days prior to the first treatment), and PEG-IFN (within 14 days prior to the first treatment). * Inability or unwillingness to give informed consent or comply with the procedures required in the protocol * Patients with household contacts who meet any of these criteria unless alternate living arrangements can be made during the patient's active dosing period and for 7 days following the last dose of study medication: * Pregnant or nursing an infant * Children \< 12 months old * History of exfoliative skin condition that at some stage has required systemic therapy * Significant immunodeficiency due to underlying illness (e.g. HIV/AIDS) and/or medication