We present a study of the molecular diffusion in freely suspended smectic-A liquid crystal films with thicknesses ranging from 20 down to only two molecular layers. The molecular mobility is directly probed by determining the trajectories of single, fluorescent tracer molecules. We demonstrate, using several different smectic compounds, that a monotonic increase of the diffusion coefficient with decreasing film thickness is a general phenomenon. In two-layer films, the diffusion is enhanced by a factor of 3 to 5 compared to the corresponding bulk smectic phase. Molecular dynamics simulations of freely suspended smectic films are presented which support the experimental results.