Recently one of us proposed a general theory of variable rest masses (VMT) compatible with post-Newtonian solar-system experiments for a wide range of its two parameters r and q, provided the asymptotic value of its fundamental field f is in a certain narrow range. Here we show that the stationary matter-free black-hole solutions of the VMT are identical to those of general relativity. In addition for r 0 (part of the range mentioned), relativistic neutron-star models in the VMT are very similar to their general-relativistic counterparts. Thus experimental discrimination between the two theories in the strong-field limit seems unfeasible. We show that in all isotropic cosmological models of the VMT capable of describing the present epoch, the Newtonian gravitational constant G/sub N/ is positive throughout the cosmological expansion. There exist nonsingular VMT cosmological solutions; this is an advantage the VMT has over general relativity. For r 0 all VMT cosmological models converge to their general-relativistic analogs at late times. As a consequence the asymptotic f attains just the required values to guarantee agreement of the VMT with post-Newtonian experiments. The VMT with r 0 allmore » VMT cosmological models converge to their general-relativistic analogs at late times. As a consequence the asymptotic f attains just the required values to guarantee agreement of the VMT with post-Newtonian experiments. The VMT with r 0 predicts a positive Nordtvedt-effect coefficient. It also predicts that G/sub N/ is currently decreasing on a time scale which could be long compared to the Hubble time. Verification of these predictions would rule out general relativity; its most natural replacement would be the VMT with r 0, and not a generic scalar-tensor theory. The success of general relativity in most respects could then be understood because the VMT with R 0 mimics it.« less