We treat the problem of combined buoyancy-thermocapillary convection in a cavity with a free surface heated differentially in the horizontal. Attention is focused on the structure and strength of the flow for large ΔT, i.e. large Marangoni and Rayleigh numbers. In the combined problem, the boundary-layer scalings for buoyant and thermocapillary convection suggest that in the limit of large ΔT, thermocapillarity will dominate the large-scale flow. Accurate numerical solutions are used to study this question at fixed cavity aspect ratio and Prandtl number, with G = Ra/Ma as a parameter. For G = 1, the flow evolves toward its boundary-layer limit in a fashion identical to that for G = 0, i.e. pure thermocapillary flow. For G = 10, the evolution is from a buoyancy-dominated structure, through a transition, to a thermocapillary-dominated structure. We infer that thermocapillarity will ultimately dominate all such flows at sufficiently large ΔT, for any fixed values of G, the aspect ratio, and the Prandtl number.