  We derive the allowed ranges of the spin--independent interaction cross
section $\sigsip$ for the elastic scattering of neutralinos on proton for wide
ranges of parameters of the general Minimal Supersymmetric Standard Model. We
investigate the effects of the lower limits on Higgs and superpartner masses
from colliders, as well as the impact of constraints from $\bsgamma$ and the
new measurement of $\gmtwo$ on the upper and lower limits on $\sigsip$. We
further explore the impact of the neutralino relic density, including
coannihilation, and of theoretical assumptions about the largest allowed values
of the supersymmetric parameters. For $\mu>0$, requiring the latter to lie
below $1\tev$ leads to $\sigsip\gsim 10^{-11}\pb$ at $\mchi\sim100\gev$ and
$\sigsip\gsim 10^{-8}\pb$ at $\mchi\sim1\tev$. When the supersymmetric
parameters are allowed above $1\tev$, for $440\gev \lsim \mchi\lsim 1020 \gev$
we derive a {\em parameter--independent lower limit} of $\sigsip \gsim 2\times
10^{-12}\pb$. (No similar lower limits can be set for $\mu<0$ nor for
$1020\gev\lsim\mchi\lsim2.6\tev$.) Requiring $\abundchi<0.3$ implies a {\em
parameter--independent upper limit} $\mchi\lsim2.6\tev$. The new $\epem$--based
measurement of $(g-2)_{\mu}$ restricts $\mchi\lsim 350\gev$ at $1 \sigma$ CL
and $\mchi\lsim515\gev$ at $2 \sigma$ CL, and implies $\mu>0$. The largest
allowed values of $\sigsip$ have already become accessible to recent
experimental searches.
