  For the vertex corrections to the partial decay rate $\Gamma(Z \to b\bar{b})$
involving the top quark only the leading terms of order $\alpha\alpha_s$ in the
$1/M_t$ expansion are known. In this work we compute the missing
next-to-leading corrections. Thus at $O(\alpha\alpha_s)$ the complete
corrections to the decay of the Z boson into bottom quarks are at hand.
