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We calculate expected event rates for direct detection of relic neutralinosas a function of parameter space of the minimal supergravity model. Numericalresults are presented for the specific case of a $^{73}$Ge detector. We findsignificant detection rates ($R> 0.01$ events/kg/day) in regions of parameterspace most favored by constraints from $B\to X_s\gamma$ and the cosmologicalrelic density of neutralinos. The detection rates are especially large inregions of large $\tan\beta$, where many conventional signals for supersymmetryat collider experiments are difficult to detect. If the parameter $\tan\beta$is large, then there is a significant probability that the first directevidence for supersymmetry could come from direct detection experiments, ratherthan from collider searches for sparticles.Comment: 25 page REVTEX file including 9 PS figure

Neutralino dark matter in minimal supergravity: Direct detection versus collider searches