  We estimate the two-loop perturbative corrections to zero-recoil matrix
elements of the flavour-changing currents $\bar c\,\gamma^\mu b$ and $\bar
c\,\gamma^\mu\gamma_5\,b$ by calculating the terms of order $n_f\,\alpha_s^2$
and substituting the dependence on the number of flavours by the first
coefficient of the $\beta$-function. Both for vector and axial vector currents,
we find moderate two-loop corrections below 1\% in magnitude. Using the
Brodsky--Lepage--Mackenzie prescription to set the scale in the
order-$\alpha_s$ corrections in the $\overline{\rm MS}$ scheme, we obtain
$\mu_V\simeq 0.92\sqrt{m_b m_c}$ and $\mu_A\simeq 0.51\sqrt{m_b m_c}$ in the
two cases. These scales are sufficiently large for perturbation theory to be
well-behaved. The implications of our results to the extraction of $|\,V_{cb}|$
are briefly discussed.
