  We consider super-gravity models in which the lightest supersymmetric
particle (LSP) is a stable gravitino. The next-to-lightest supersymmetric
particle (NLSP) freezes out with its thermal relic density and then decays
after $(10^5-10^{10})$ sec, injecting high-energy photons into the cosmic
plasma. These photons heat up the electron plasma which then thermalizes with
the cosmic microwave background (CMB) via Compton scattering, bremsstrahlung
and double-Compton scattering. Contrary to previous studies which assume
instantaneous energy injection, we solve the full kinetic equation for the
photon number density with a source term describing the decay of the NLSP. This
source term is based on the requirement that the injected energy be almost
instantaneously redistributed by Compton scattering, hence leading to a
time-dependent chemical potential. We investigate the case of a stau NLSP and
determine the constraints on the gravitino and stau masses from observations of
the CMB spectrum by assuming that all gravitino LSPs come from stau NLSP
decays. Unlike the analytical approximations, we find that there may be a stau
mass below which the constraint from the CMB spectrum vanishes.
