At Taylor Glacier, a cold-based outlet glacier of the East Antarctic ice sheet, observed surface speeds in the terminus region are 20 times greater than those predicted using Glen’s flow law for cold (–17°C), thin (100 m) ice. Rheological properties of the clean meteoric glacier ice and the underlying deformable debris-rich basal ice can be inferred from surface-velocity and ablation-rate profiles using inverse theory. Here, with limited data, we use a two-layer flowband model to examine two end-member assumptions about the basal-ice properties: (1) uniform softness with spatially variable thickness and (2) uniform thickness with spatially variable softness. We find that the basal ice contributes 85–98% to the observed surface velocity in the terminus region. We also find that the basal-ice layer must be 10–15 m thick and 20–40 times softer than clean Holocene-age glacier ice in order to match the observations. Because significant deformation occurs in the basal ice, our inverse problem is not sensitive to variations in the softness of the meteoric ice. Our results suggest that despite low temperatures, highly deformable basal ice may dominate flow of cold-based glaciers and rheologically distinct layers should be incorporated in models of polar-glacier flow.