We present a theoretical study of the evolution of a drop of pure liquid on a solid substrate, which it wets completely. In a situation where evaporation is significant, the drop does not spread, but instead the drop radius goes to zero in finite time. Our description couples the viscous flow problem to a self-consistent thermodynamic description of evaporation from the drop and its precursor film. The evaporation rate is limited by the diffusion of vapor into the surrounding atmosphere. For flat drops, we compute the evaporation rate as a nonlocal integral operator of the drop shape. Together with a lubrication description of the flow, this permits an efficient numerical description of the final stages of the evaporation problem. We find that the drop radius goes to zero like R∝(t0−t)α, where α has value close to 1/2, in agreement with experiment.