The evaporation of a small droplet of a volatile fluid from a solid surface differs according to whether the fluid wets the surface. When the initial contact angle is less than 90 degrees, the evaporation initially proceeds with an accompanying decrease in the contact angle but no change in the contact radius, This stage of evaporation dominates the time scale and has previously been described by a diffusion model. However, when the initial contact angle is greater than 90 degrees, it is the contact radius that decreases rather than the contact angle. In this work we report detailed measurements of the evaporation of drops of water fi om Tenon film. On deposition the contact angles rapidly relax from around 112 degrees to 108 degrees, which is the equilibrium value for droplets in a saturated vapor. The angle then remains constant for the majority of the evaporation time, Eventually the system changes to a mode of evaporation in which both contact angle and radius change simultaneously. The data are compared to an extension of the diffusion model. and this provides estimates of the diffusion coefficient. It is suggested that the constant contact angle observed during much of the evaporation is a result of a local saturation of the vapor in the region of the contact line.