Weathering of rock surfaces is often associated with a surface dissolution process. Chemical interactions occur on grain boundaries and diffusion is the controlling process. A dissolution boundary layer (rind) develops adjacent to the weathering surface. We quantify the extent of dissolution by introducing a damage variable f, f = 0 for pristine rock, and when f = f0 the rock disintegrates. We assume that the variations of the damage variable are given by the diffusion equation. We solve two problems. The first is for the structure of the transient dissolution boundary layer prior to surface disintegration. We find an incubation time ti before active weathering (disintegration) begins. The second is the solution for steady state weathering with a constant weathering velocity vw. Our results are entirely consistent with weathering studies on Carrera marble gravestones in the United Kingdom.