If the Sun contained even a small mass fraction of weakly interacting massive particles, there could be significant effects on central solar structure. The long mean free paths associated with such particles make them very efficient energy conductors. Consequently, an essentially isothermal core can be produced, which removes the central temperature peak responsible for the bulk of the predicted solar neutrinos in Davis's /sup 37/Cl neutrino capture experiment. We explore the solar evolutionary consequences of both imposed isothermal core models and models including energy transport by a specific class of weakly interacting particles. In particular, a relative mass fraction, 10/sup -8/, of 4GeV particles reduces predicted neutrino count rates by more than a factor of 3.