  We make a detailed study on the $J/\psi$ meson longitudinal leading-twist
distribution amplitude $\phi_{2;J/\psi}^\|$ by using the QCD sum rules within
the background field theory. By keeping all the non-perturbative condensates up
to dimension-six, we obtain accurate QCD sum rules for the moments
$\langle\xi_{n;J/\psi}^\|\rangle$. The first three ones are
$\langle\xi_{2;J/\psi}^\|\rangle=0.083(12)$,
$\langle\xi_{4;J/\psi}^\|\rangle=0.015(5)$ and
$\langle\xi_{6;J/\psi}^\|\rangle=0.003(2)$, leading to a single peaked behavior
for $\phi_{2;J/\psi}^\|$ which is sharper than the previous ones around the
region of $x\sim0.5$. As an application, we adopt the QCD light-cone sum rules
to calculate the $B_c$ meson semileptonic decay $B_c^+ \to J/\psi \ell^+
\nu_\ell$. We obtain $\Gamma(B_c^+ \to J/\psi \ell^+ \nu_\ell) =
(89.67^{+24.76}_{-19.06}) \times 10^{-15}~{\rm GeV}$ and $\Re(J/\psi \ell^+
\nu_\ell) = 0.217^{+0.069}_{-0.057}$, which agree with the next-to-leading
order pQCD prediction and the new CDF measurement within errors.
