  We present the results of the complete one-loop computation of the
$B^0_{d,s}\to l^+l^-$ decay rate in the MSSM. Both sources of the FCNC, the CKM
matrix and off-diagonal entries of the sfermion mass matrices are considered.
Strong enhancement of the branching ratio (compared to the SM prediction) can
be obtained in the large $\tan\beta\sim m_t/m_b$ regime in which the neutral
Higgs boson ``penguin'' diagrams dominate. We make explicit the strong
dependence of this enhancement on the top squarks mixing angle in the case of
the chargino contribution and on the $\mu$ parameter in the case of the gluino
contribution. We show that, in some regions of the MSSM parametre space, the
branching ratio for this process can be as large as $10^{-(5-4)}$ respecting
all existing constraints, including the CLEO measurement of $BR(B\to
X_s\gamma)$. We also estimate, that for chargino and stop masses $\sim{\cal
O}(100$ GeV) $BR(B^0_s\to l^+l^{\prime -})$ with $ll^\prime=e\tau$ or $\mu\tau$
can be of the order of $10^{-11}$ for the still allowed values of the
off-diagonal entries in the slepton mass matrix.
