  We have recently seen new upper bounds for $B^0_s\to \mu^+\mu^-$, a key decay
to search for physics beyond the Standard Model. Furthermore a non-vanishing
decay width difference $\Delta\Gamma_s$ of the $B_s$ system has been measured.
We show that $\Delta\Gamma_s$ affects the extraction of the $B^0_s\to
\mu^+\mu^-$ branching ratio and the resulting constraints on the New Physics
parameter space, and give formulae for including this effect. Moreover, we
point out that $\Delta\Gamma_s$ provides a new observable, the effective
$B^0_s\to \mu^+\mu^-$ lifetime $\tau_{\mu^+\mu^-}$, which offers a
theoretically clean probe for New Physics searches that is complementary to the
branching ratio. Should the $B^0_s\to \mu^+\mu^-$ branching ratio agree with
the Standard Model, the measurement of $\tau_{\mu^+\mu^-}$, which appears
feasible at upgrades of the LHC experiments, may still reveal large New Physics
effects.
