Inclusive decay of B mesons into $D_s$ or $D^*_s$

We compute the inclusive decay rates $b \to D_s^-(D^{*-}_s) c$ including lowest order QCD corrections on the quark legs, and compare with existing data. Unlike the short distance QCD corrections, that are of higher order, these corrections are of order $\alpha_s$. In the on-shell renormalization scheme and for $\alpha_s(m_b) \cong 0.2$ we find a correction of -10% to the inclusive rate computed using factorization. This gives a total rate $BR(b \to D_s^-(D^{*-}_s)c) \cong 8%$ consistent within $1\sigma$ with the measured value $BR(B \to D_s^{\pm}X) = (10.0 \pm 2.5)%$. The general formulae given here include the case of vanishing mass for the final quark $b \to D_s^-(D_s^{*-})u$. The radiative correction to this rate is -17%. We show in another place that this process can be useful for the measurement of the CKM matrix element $V_{ub}$. We also give the renormalized vertex at the interesting values $q^2 = 0$ and $q^2 = q_{max}^2$, and compare with existing literature.