The Mikheyev-Smirnov-Wolfenstein (MSW) resonant neutrino oscillation mechanism is investigated for the Sun using a detailed numerical solar model and a modified version of the Parke-Walker technique for following the neutrino phases through the oscillation resonance. We present overall solar-neutrino spectra and the associated expected neutrino count rates for the Cl37, Ga71, and Kamiokande detectors for ranges of masses and vacuum mixing angles for two neutrino species. We also investigate the effects of uncertainties in the solar model. In particular, we examine the effect of opacity changes on the expected solar-neutrino spectrum and resulting parameter space for the MSW mechanism. We find that plausible uncertainties in the standard sol...