  A search for the rare decays $B^0_s\to\mu^+\mu^-$ and $B^0\to\mu^+\mu^-$ is
performed at the LHCb experiment using data collected in $pp$ collisions
corresponding to a total integrated luminosity of 4.4 fb$^{-1}$. An excess of
$B^0_s\to\mu^+\mu^-$ decays is observed with a significance of 7.8 standard
deviations, representing the first observation of this decay in a single
experiment. The branching fraction is measured to be ${\cal
B}(B^0_s\to\mu^+\mu^-)=\left(3.0\pm 0.6^{+0.3}_{-0.2}\right)\times 10^{-9}$,
where the first uncertainty is statistical and the second systematic. The first
measurement of the $B^0_s\to\mu^+\mu^-$ effective lifetime,
$\tau(B^0_s\to\mu^+\mu^-)=2.04\pm 0.44\pm 0.05$ ps, is reported. No significant
excess of $B^0\to\mu^+\mu^-$ decays is found and a 95 % confidence level upper
limit, ${\cal B}(B^0\to\mu^+\mu^-)<3.4\times 10^{-10}$, is determined. All
results are in agreement with the Standard Model expectations.
