  A measurement of the branching ratio for the rare decay mode B_s->mu mu at
the Tevatron is an opportunity to test various supersymmetric scenarios. We
investigate the prospects for studying this mode in Run II and estimate that
CDF would be sensitive to this decay for a branching ratio > 1.2 \times 10^{-8}
with 15 \invfb (or, if a similar analysis holds for D0, >6.5\times 10^{-9} for
the combined data). We calculate the branching ratio in minimal supergravity
(mSUGRA) parameter space, and find that tan\beta > 30 can be probed. (This
mSUGRA parameter space cannot be probed by direct production of SUSY particles
at Run II.) Including other experimental constraints on the mSUGRA parameter
space, one finds that CDF \bsmumu measurements would be able to cover the full
mSUGRA parameter space for tan\beta = 50 if the muon g_{\mu} - 2 anomaly
exceeds \sim 11 \times 10^{-10}, and about half the allowed parameter space for
tan\beta = 40. A large branching ratio > 7(14) \times 10^{-8} (feasible with
only 2 \invfb) would be sufficient to exclude the mSUGRA model for tan\beta\leq
50(55). Dark matter neutralino-proton detection cross sections are examined in
the allowed region, and should be large enough to be accessible to future
planned experiments. Combined measurements of \bsmumu the Higgs mass m_h and
the muon g_{\mu}-2 anomaly would be sufficient to determine the \mu>0 mSUGRA
parameters (or show the model is inconsistent with the data). We also briefly
discuss the \bsmumu decay in R parity violating models. There, for some models,
the branching ratio can be large enough to be detected even for small tan\beta
and large \mhalf.
