  We compute the QCD corrections to the decay rate difference in the $B_s-\bar
B_s$ system, $\Delta\Gamma_{B_s}$, in the next-to-leading logarithmic
approximation using the heavy quark expansion approach. Going beyond leading
order in QCD is essential to obtain a proper matching of the Wilson
coefficients to the matrix elements of local operators from lattice gauge
theory. The lifetime difference is reduced considerably at next-to-leading
order. We find $(\Delta\Gamma/\Gamma)_{B_s}=(f_{B_s}/210 MeV)^2 [0.006
B(m_b)+0.150 B_S(m_b)-0.063]$ in terms of the bag parameters $B, B_S$ in the
NDR scheme. As a further application of our analysis we also derive the
next-to-leading order result for the mixing-induced CP asymmetry in inclusive
$b\to u\bar ud$ decays, which measures $\sin 2\alpha$.
