The dissolution of a gas bubble in a viscous Newtonian liquid in simple shear and creeping flow was studied experimentally. A simple theoretical model for bubble growth or dissolution at high Peclet numbers is also presented. The main result of the model shows that for high capillary numbers (slender bubbles) the dissolution or growth rare is proportional to the shear rate to the 3/8 power. The experimental work was carried out in a Couette apparatus. Air and carbon dioxide bubbles were injected into polymer liquids (low-molecular-weight polyisobutylenes) and photographed under shear. No "tip-streaming" was observed, and experimental results compared favorably with predictions of the simple model, based on a mechanism that combines diffusion and convection.