We have studied the flow of Carbopol, a polymer microgel, confined to approximately square microchannels with widths ranging from 451 mu m down to 54 mu m. Velocity profiles in the midplane of the channels were measured using particle image velocimetry. Carbopol is a yield-stress fluid, and the velocity profiles measured in the larger channels agreed well with simulations based on its measured bulk-scale rheology. In microchannels 150 mu m or smaller in width, however, the velocity profiles could not be fitted by a model with a finite yield stress, but instead were well-described by a power-law model with zero yield stress. We explain the vanishing of the yield stress in terms of the confinement by the microchannels of structural elements in the Carbopol.