Is SUSY accessible by direct dark matter detection?

We performed a combined analysis of the parameter space of the Minimal Supersymmetric Standard Model (MSSM) taking into account cosmological and accelerator constraints including those from the radiative $b\rightarrow s\gamma$ decay measured by the CLEO collaboration. Special attention is paid to the event rate, $R$, of direct dark matter neutralino detection. We have found domains of the parameter space with $R\simeq 5-10$~events/kg/day. This would be within the reach of current dark matter experiments. The $b\rightarrow s\gamma$ data do not essentially reduce these large event rate domains of the MSSM parameter space.