  We complete the QCD sum rule analysis of the Isgur Wise form factor
$\xi(v\cdot v')$ at next-to-leading order in renormalization-group improved
perturbation theory. To this end, the exact result for the two-loop corrections
to the perturbative contribution is derived using the heavy quark effective
theory. Several techniques for the evaluation of two-loop integrals involving
two different types of heavy quark propagators are discussed in detail, among
them the methods of integration by parts and differential equations. The
order-$\alpha_s$ corrections to the Isgur-Wise function turn out to be small
and well under control. At large recoil, they tend to decrease the form factor
by $5-10\%$.
