  We derive new sum rules for the form factors of the $B_c\to J/\psi\, \ell
\bar{\nu}_\ell$ semileptonic transitions, employing the vacuum-to-$B_c$
correlation function of the $J/\psi$-interpolating and $b\to c$ weak currents.
In the heavy quark limit and at a space-like momentum transfer to the weak
current, a local operator-product expansion is valid for this correlation
function. As a result, in the leading power, the non-perturbative input is
reduced to the decay constant of $B_c$ meson. Furthermore, applying hadronic
dispersion relation in the $J/\psi$ channel, we find that a non-vanishing OPE
spectral density in the duality interval of $J/\psi$ emerges only at
$\mathcal{O}(\alpha_s)$. We calculate this density in the relevant kinematic
regime. The $B_c\to J/\psi$ form factors at space-like momentum transfer are
then calculated from the new sum rules.
