Mikheyev and Smirnov have observed that neutrino oscillations in the Sun can be greatly enhanced through the mechanism of Wolfenstein matter oscillations. We develop a qualitative understanding of this phenomenon in the small-mixing-angle limit and carry out extensive calculations in order to apply it to the solar-neutrino problem. Our simple theoretical model agrees remarkably well with the calculations. After determining those values of ..delta..m-italic/sup 2/ and sin/sup 2/2theta in the small-mixing-angle limit for which the /sup 8/B plus /sup 7/Be neutrino capture rate in /sup 37/Cl is suppressed by a factor 2--4, we predict the corresponding capture rate for p-italicp-italic plus /sup 7/Be neutrinos in /sup 71/Ga. The gallium capture rate can range from no reduction to a factor of 10 reduction. We also determine the modified spectrum of /sup 8/B neutrinos arriving at Earth and discuss the importance of this spectrum as a means of choosing between oscillations and the solar model as the cause of the solar-neutrino problem, and also as a means of distinguishing between different sets of oscillation parameters.