  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, $|\tau(B_s)/\tau(B_d)-1|<1\%$.
