A new, simple method for treating the short-range correlations in a one-positron, many-electron system is proposed. The method puts the emphasis on a real-space representation of wave functions. Our ultimate goal is to apply this method to situations in which the electron density is nonuniform, such as a metallic surface. In the present paper, we concentrate on the case of a uniform electron gas, which provides a useful test of the method. Our results for the total annihilation rate are well behaved at all densities, tend to the positronium limit at low density, and agree with experiment for simple metals. We show that the total rate is insensitive to electron-electron correlations, although such correlations could affect the momentum dependence of the partial annihilation rate measured in angular-correlation experiments. The extension of the method to nonuniform systems is briefly sketched.