The purpose of this study is to determine the feasibility and safety of using small beads (70-150 micron in place of 100-300 micron) to deliver chemotherapy into the liver to treat patients with hepatocellular carcinoma (HCC). The beads (LC-Bead M1) will be loaded with doxorubicin (DEBDOX-M1), and used to administer transarterial chemoembolization (TACE) DEBDOX, loaded with doxorubicin, is a device that utilizes tiny beads (70-150 microns) to deliver chemotherapy agents into liver tumor(s) via the hepatic artery. This device allows for continuous release of doxorubicin into the liver tumor tissue(s) causing necrosis of the targeted tumor(s). The potential advantages of the smaller beads are deeper penetration into the tumor bed, while avoiding premature proximal occlusion of vessels feeding the tumor, and more consistent dosing. Response to therapy will be evaluated monthly by clinic visits and blood tests (to include assessment of liver function and tumor markers) and by imaging (usually MRIs) every 1-2 months. Patients will be on study for 6 months after which they will be exited from the study and followed for survival. Once exited from the study they will continue to be eligible to receive the smaller beads (DEBDOX), should it be recommended.