The Born-Kirkwood-Onsager (BKO) model of solvation, where a solute molecule is positioned inside a cavity cut into a solvent, which is considered as a dielectric continuum, is studied within the bounds of nonlocal electrostatics. The nonlocal cavity model is explicitly formulated and the corresponding nonlocal Poisson equation is reduced to an integral equation describing the behavior of the charge density induced in the medium. It is found that the presence of a cavity does not create singularities in the total electrostatic potential and its normal derivatives. Such singularities appear only in the local limit and are completely dissipated by nonlocal effects. The Born case of a spherical cavity with a point charge at its centre is investigated in detail. The corresponding one-dimensional integral Poisson equation is solved numerically and values for the solvation energy are determined. Several tests of this approach are presented : (a) We show that our integral equation reduces in the local limit to the chief equation of the local BKO theory. (b) We provide certain approximations which enable us to obtain the solution corresponding to the preceding nonlocal treatment of Dogonadze and Kornyshev (DK). (c) We make a comparison with the results of molecular solvation theory (mean spherical approximation), as applied to the calculation of solvation energies of spherical ions.