  The two-loop QCD radiative corrections to the $B_c$ meson leptonic decay rate
are calculated in the framework of NRQCD factorization formalism. Two types of
master integrals appearing in the calculation are obtained analytically for the
first time. We get the short-distance coefficient of the leading matrix element
to order $\alpha_s^2$ by matching the full perturbative QCD calculation results
to the corresponding NRQCD ones. The result in this work helps the evaluation
of the $B_c$ leptonic decay constant, as well as the Cabibbo-Kobayashi-Maskawa
matrix element $|V_{cb}|$, to the full next-to-next-to-leading order degree of
accuracy.
