We use the heavy quark expansion to investigate the width difference
$\Delta\Gamma_{B_s}$ between the $B_s$ mass eigenstates. The
corrections of ${\cal
O}(\Lambda_{QCD}/m_b)$ and ${\cal O}(m_s/m_b)$ to the leading order
expression
in the operator product expansion are derived and estimated to yield
a sizable reduction
of the leading result for $\Delta\Gamma_{B_s}$ by typically $30\%$.
For completeness
we also quantify small effects due to penguin operators and CKM
suppressed
contributions. Based on our results we discuss the prediction for
$(\Delta\Gamma/\Gamma)_{B_s}$ with particular emphasis on theoretical
uncertainties.
We find $(\Delta\Gamma/\Gamma)_{B_s}=0.16^{+0.11}_{-0.09}$, where the
large
error is dominated by the uncertainty in hadronic matrix elements. An
accuracy of about
$10\%$ in $(\Delta\Gamma/\Gamma)_{B_s}$ should be within reach,
assuming
continuing progress in lattice calculations. In addition we address
phenomenological
issues and implications of a $\Delta\Gamma_{B_s}$ measurement for
constraints on
$\Delta M_{B_s}$ and CKM parameters. We further consider in some
detail the
lifetime ratio $\tau(B_s)/\tau(B_d)$ and estimate that, most likely,
$|\tau(B_s)/\tau(B_d)-1|<1\%$.