  We compute the scattering rates for the lightest neutralino $\chi^0_1$ in the
forthcoming germanium $(^{73}{\rm Ge}+^{76}{\rm Ge})$ detector and a proposed
lead detector $(^{207}{\rm Pb})$, within the framework of $SU(5)\times U(1)$
supergravity. We find that in only a small portion ($\lsim10\%$) of the
parameter spaces of this class of models, are the rates in the germanium
detector above the expected initial experimental sensitivity of 0.1
events/kg/day. However, a much larger portion ($\lsim40\%$) of the parameter
spaces could be probed with an improved background rejection capability (0.01
events/kg/day) and/or a more sensitive detector $(^{207}{\rm Pb})$.
