  The QCD radiative corrections to the leptonic decay rate of the $B_c$ meson
are calculated using the formalism of nonrelativistic QCD (NRQCD) to separate
short-distance and long-distance effects. The $B_c$ decay constant is factored
into a sum of NRQCD matrix elements each multiplied by a short-distance
coefficient. The short-distance coefficient for the leading matrix element is
calculated to order $\alpha_s$ by matching a perturbative calculation in full
QCD with the corresponding perturbative calculation in NRQCD. This
short-distance correction decreases the leptonic decay rate by approximately
$15\%$.
