A theory is presented to quantify the electrostatic forces in thin aqueous electrolyte films between two charge regulating surfaces. The Poisson-Boltzmann equation, as modified by Gur et al. (J. Colloid Interface Sci. 64, 326 (1978)) to include the effects of hydration free energy of ions, is numerically solved for overlapping double layers. It is shown that the reduction in dielectric constant with increasing electric field increases the hydration free energy of the ions in the double layer, giving rise to an additional repulsive force. These "hydration" forces follow an exponential or a power-law relation with film thickness depending upon the degree of dielectric saturation in the double layer. The results obtained agree remarkably well with the experimental data of Pashley (J. Colloid Interface Sci. 80, 153 (1981)) at low electrolyte concentration and pH. At higher concentration and pH, the model provides qualitative agreement with experimental observations. It is concluded that dielectric saturation of the double layer near highly charged interfaces may play an important role in electrostatic interactions in thin electrolyte films.