The modification of the electrostatic continuum solvent model considered in the present work is based on the exact solution of the Poisson equation, which can be constructed provided that the dielectric permittivity epsilon of the total solute and solvent system is an isotropic and continuous spatial function. This assumption allows one to formulate a numerically efficient and universal computational scheme that covers the important case of a variable epsilon function inherent to the solvent region. The obtained type of solution is unavailable for conventional dielectric continuum models such as the Onsager and Kirkwood models for spherical cavities and the polarizable continuum model (PCM) for solute cavities of general shape, which imply that epsilon is discontinuous on the boundary confining the excluded volume cavity of the solute particle. Test computations based on the present algorithm are performed for water and several nonaqueous solvents. They illustrate specific features of this approach, called the "smooth boundary continuum model" (SBCM), as compared to the PCM procedure, and suggest primary tentative results of its parametrization for different solvents. The calculation for the case of a binary solvent mixture with variable epsilon in the solvent space region demonstrates the applicability of this approach to a novel application field covered by the SBCM.