Measurements are presented of the long-time self-diffusion coefficient and of the low shear limiting viscosity of dispersions of charge stabilized colloidal silica spheres. Long-time self-diffusion coefficients were measured using fluorescence recovery after photobleaching (FRAP), the theory of which is presented and generalized to Gaussian laser beams. The particles, suspended in solutions of LiCI in dimethylformamide, interacted via a screened Coulomb potential, the range of which was varied through the ionic strength. Measurements were made up to volume fractions beyond freezing where a coexistence occurred between a colloidal crystal and a colloidal fluid. It is often speculated that the long-time self-diffusion coefficient and the low shear viscosity of a dispersion are related through a simple Stokes-Einstein-like relation, but this expectation is not confirmed by the experiments. A slightly modified relation, however, does seem to provide a reasonable empirical description of the data.