We study the evaporating meniscus of a perfectly wetting liquid formed in the gap between two horizontal flat plates. The plates are at the same temperature as the surroundings, and the pure liquid evaporates into a binary mixture of its own vapor and an inert component. Though the identical problem for a capillary has been studied since the work of Stefan, except for the work by Derjaguin et al. (1965), the interface has been treated as a plane surface having a known contact angle of 90 degrees. Here, by contrast, the interface location is determined as part of the solution to a free boundary problem coupling viscous flow in the liquid to diffusive transport in the gas. We make the following simplifying assumptions: (a) liquid and vapor at the interface are in local thermodynamic equilibrium, and (b) the system is effectively isothermal. Given (a) and (b), the vapor partial pressure is related to the liquid pressure by the Kelvin equation. Though the hydrostatic contact angle is zero for a completely wetting system, the stationary evaporating meniscus exhibits an apparent contact angle; is determined chiefly by a capillary number Ca=V-s/ based on surface tension , liquid viscosity , and a velocity scale V-s set by evaporation. Though microphysics must be included in the free boundary problem in order to resolve a hydrodynamic singularity at the contact line, is insensitive to the microphysical details.