  The direct detection of neutralino dark matter is analysed in general
supergravity scenarios, where non-universal soft scalar and gaugino masses can
be present. In particular, the theoretical predictions for the
neutralino-nucleon cross section are studied and compared with the sensitivity
of dark matter detectors. We take into account the most recent astrophysical
and experimental constraints on the parameter space, including the current
limit on B(Bs-> mu+ mu-). The latter puts severe limitations on the dark matter
scattering cross section, ruling out most of the regions that would be within
the reach of present experiments. We show how this constraint can be softened
with the help of appropriate choices of non-universal parameters which increase
the Higgsino composition of the lightest neutralino and minimise the chargino
contribution to the b->s transition.
