  We calculate dark matter scattering rates in the minimal supersymmetric
extension of the Standard Model (MSSM), allowing the soft
supersymmetry-breaking masses of the Higgs multiplets, m_{1,2}, to be
non-universal (NUHM). Compared with the constrained MSSM (CMSSM) in which
m_{1,2} are required to be equal to the soft supersymmetry-breaking masses m_0
of the squark and slepton masses, we find that the elastic scattering cross
sections may be up to two orders of magnitude larger than values in the CMSSM
for similar LSP masses. We find the following preferred ranges for the
spin-independent cross section: 10^{-6} pb \ga \sigma_{SI} \ga 10^{-10} pb, and
for the spin-dependent cross section: 10^{-3} pb \ga \sigma_{SD}, with the
lower bound on \sigma_{SI} dependent on using the putative constraint from the
muon anomalous magnetic moment. We stress the importance of incorporating
accelerator and dark matter constraints in restricting the NUHM parameter
space, and also of requiring that no undesirable vacuum appear below the GUT
scale. In particular, values of the spin-independent cross section another
order of magnitude larger would appear to be allowed, for small \tan \beta, if
the GUT vacuum stability requirement were relaxed, and much lower cross-section
values would be permitted if the muon anomalous magnetic moment constraint were
dropped.
