  We compute the genuine SUSY one-loop quantum contributions to
flavour-changing MSSM Higgs-boson decays into $b \bar s$ and $s \bar b$ using
the full diagrammatic approach that is valid for all $\tan \beta$ values and do
not rely on the mass-insertion approximation for the characteristic
flavour-changing parameter. We analyze in full detail the dependence of these
flavour-changing partial widths on all the relevant MSSM parameters and also
study the non-decoupling behaviour of these widths with the SUSY mass
parameters. We find that these contributions are sizable as compared to the SM
ones, and can be very efficient as an indirect method in the future search for
Supersymmetry.
