We determine the static dielectric function of the electron gas at selected densities by using variational and fixed-node diffusion quantum Monte Carlo to find the response to a sinusoidal electric field. The dielectric function is then calculated by relating the induced charge density to the change in ground-state energy as a function of the amplitude of the applied field. A released-node Green's-function Monte Carlo calculation shows that the diffusion results are exact within the determined error bars, apart from the size dependence correction for extrapolation to the thermodynamic limit. The dielectric response is consistent at long wavelengths with the compressibility sum rule derived from a fit to Ceperley and Alder's Green's-function Monte Carlo results, but exhibits dfferences from accepted dielectric models at intermediate wave vectors.