  The requirement that the lightest neutralino $\tilde\chi_1^0$ has the right
thermal relic density to explain all Dark Matter in the universe strongly
constrains the parameter space of supersymmetric models in general, and of the
mSUGRA model in particular. Recently improved calculations of the mass of the
light CP-even Higgs boson $h$ present in this model, and the increased central
value of the mass of the top quark, have re--opened the possibility that $2
\mlsp \lsim m_h$. In this ``$h-$pole region'' the LSP annihilation cross
section is enhanced by near-resonant $h$ exchange in the $s-$channel, reducing
the relic density to acceptable values. We delineate the corresponding region
of mSUGRA parameter space, and explore its phenomenology. In particular, we
find strong upper bounds on the masses of the gluino, lighter chargino and LSP.
