  We calculate expected event rates for direct detection of relic neutralinos
as a function of parameter space of the minimal supergravity model. Numerical
results are presented for the specific case of a $^{73}$Ge detector. We find
significant detection rates ($R> 0.01$ events/kg/day) in regions of parameter
space most favored by constraints from $B\to X_s\gamma$ and the cosmological
relic density of neutralinos. The detection rates are especially large in
regions of large $\tan\beta$, where many conventional signals for supersymmetry
at collider experiments are difficult to detect. If the parameter $\tan\beta$
is large, then there is a significant probability that the first direct
evidence for supersymmetry could come from direct detection experiments, rather
than from collider searches for sparticles.
