The rates of photoionization and photodissociation of molecules are calculated as functions of visual extinction into plane-parallel clouds illuminated by the average interstellar radiation field. An accurate technique is used to calculate the attenuation of the ultraviolet radiation due to scattering and absorption by dust grains inside the clouds. As a plausible and self-consistent model for the grains in diffuse clouds and the outer layers of dense clouds, the graphite-silicate grain mixture of Mathis, Rumpl, & Nordsieck is adopted, together with the wave-length-dependent grain optical properties calculated by Draine & Lee. Depth-dependent photodissociation and photoionization rates are presented for a wide range of molecules, and the results are compared with the rates of photodestruction resulting from internal photons generated by cosmic rays.