  We examine the cross sections for the elastic scattering of neutralinos
$\chi$ on nucleons $p,n$, as functions of $m_\chi$ in the constrained minimal
supersymmetric standard model. We find narrow bands of possible values of the
cross section, that are considerably lower than some previous estimates. The
constrained model is based on the minimal supergravity-inspired framework for
the MSSM, with universal scalar and gaugino masses $m_0, m_{1/2}$, and $\mu$
and the MSSM Higgs masses treated as dependent parameters. We explore
systematically the region of the $(m_{1/2}, m_0)$ plane where LEP and other
accelerator constraints are respected, and the relic neutralino density lies in
the range $0.1 \le \Omega_{\chi} h^2 \le 0.3$ preferred by cosmology. We update
previous discussions of both the spin-independent and -dependent scattering
matrix elements on protons and neutrons, using recent analyses of low-energy
hadron experiments.
