It is proposed that the evaporation of black holes via Hawking radiation may conclude at densities and temperatures no higher than those invoked during the inflationary stage of common cosmological models, thus avoiding the need for quantum gravity or physics at the Planck length in the analysis. The false vacuum associated with cosmological inflation might produce sufficient antigravity to halt the approach to a singularity inside black holes, as matter there is compressed to conditions where a phase transition to false vacuum should occur. This conjecture is supported by a very simplified spherical model of a stationary non-singular black hole containing some false vacuum in a central part of the spacetime well inside the Schwarzschild event horizon.