TRC105 for Liver Cancer That Has Not Responded to Sorafenib

* INCLUSION CRITERIA: * Patients must have histopathological confirmation of Hepatocellular Carcinoma (HCC) by the Laboratory of Pathology of the National Cancer Institute (NCI) prior to entering this study. Or histopathological confirmation of carcinoma in the setting of clinical and radiological characteristics which, together with the pathology, are highly suggestive of a diagnosis of HCC. * Patients must have disease that is not amenable to potentially curative resection or ablative techniques. In addition, disease must not be amenable to transhepatic arterial chemoembolization (TACE). Patients may have had prior TACE and had disease progression following it. Patients must not be considered potential candidates for liver transplantation. This determination will be made after hepatobiliary surgical input at the NCI multidisciplinary conference. * All patients enrolled will be required to have measurable disease. * If liver cirrhosis is present, patient must have a Child-Pugh A classification. * Patients must have progressed on or been intolerant of prior sorafenib therapy. * Patients must with cirrhosis have had esophagogastric endoscopy within the previous 6 months prior to study entry for the assessment of varices. If the patient has not had this done they must be willing to undergo this procedure prior to study entry. * Age greater than or equal to 18 years * Life expectancy of greater than 3 months. * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 * Patients must have normal organ and marrow function as defined below: * Absolute neutrophil count greater than or equal to 1,500/mcL * Platelets greater than or equal to 60,000/mcL * Total bilirubin less than or equal to 3 times upper normal limits * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) less than or equal to 10 times upper limit of normal * Creatinine less than or equal to 1.5 times upper normal limits OR creatinine clearance greater than or equal to 40 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal, as calculated by the Cockcroft Gault formula. * Patients must have recovered from any acute toxicity related to prior therapy, including surgery. Toxicity should be less than or equal to grade 1 or returned to baseline. * Patients must not have other invasive malignancies within the past 3 years (with the exception of non-melanoma skin cancers, carcinoma in situ of the cervix and noninvasive bladder cancer). * Enrolled patients must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, the duration of study participation and 3 months after the end of the treatment. * Patient must be able to understand and willing to sign a written informed consent document. EXCLUSION CRITERIA: * Patients who have had chemotherapy, large field radiotherapy, or major surgery must wait 4 weeks prior to entering the study (2 weeks is sufficient for targeted systemic therapy provided toxicity has recovered to less than or equal to grade 1). * Patients may not be receiving any anti-cancer agents not approved by the Food and Drug Administration (FDA) within the past 4 weeks. * Patients with known brain metastases will be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. * Proteinuria, as demonstrated by a 24 hour protein of greater than or equal to 2000 mg. Urine protein will be screened by urine protein-creatinine ratio (UPC). For urine protein-to-creatinine (UPC) ratio \> 1.0, a 24-hour urine protein will need to be obtained and the level should be \< 2000 mg for patient enrollment. * Uncontrolled intercurrent illness including, but not limited to, hypertension (systolic blood pressure (BP) 160, diastolic BP \> 100), ongoing or active systemic infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia or psychiatric illness/social situations that would limit compliance with study requirements. * No anti-coagulation therapy is allowed with the exception of low-dose aspirin. * No bleeding diathesis. * Patients with a history of bleeding varices in previous 1 year are excluded (unless patient has subsequently had a liver transplant). Those with gastric varices or varices that are deemed as high risk by the endoscopist should be placed on appropriate medical therapy as advised by the gastroenterologist. * History of peptic ulcer disease or hemorrhagic gastritis within 6 months of TRC105 administration, unless patient has received adequate treatment for peptic ulcer disease or hemorrhagic gastritis and has evidence of complete resolution documented by esophagogastroduodenoscopy (EGD). Mild gastritis is allowed. * Corrected QT interval (QTc) \> 500 msec. * Human immunodeficiency virus (HIV)-positive patients receiving anti-retroviral therapy are excluded from this study due to the possibility of pharmacokinetic interactions between antiretroviral medications and TRC105. HIV positive patients not receiving antiretroviral therapy are excluded due to the possibility that TRC105 may worsen their condition and the likelihood that the underlying condition may obscure the attribution of adverse events with respect to TRC105. * History of hypersensitivity reaction to human or mouse antibody products. * Patients with a history of familial bleeding disorders. * Patients with a history of hereditary hemorrhagic telangiectasia (Osler-Weber-Rendu syndrome). * Pregnancy and breast feeding are exclusion factors. Enrolled patients must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, the duration of study participation and 3 months after the end of the treatment. INCLUSION OF WOMEN AND MINORITIES: -Men and women of all races and ethnic groups are eligible for this trial.