  We calculate the cross section of the exclusive process $e^{+} +e^{-}\to
J/\psi+\eta_c$ at the leading order approximation within the QCD light-cone sum
rules (LCSR) approach. It is found that the form factor
$F_{VP}(V=J/\psi,P=\eta_c)$ depends mainly on the behavior of the twist-2
distribution amplitude of the $\eta_c$-meson at the scale of this process. Thus
in order to obtain a reliable estimation of the cross section, it is important
to have a realistic distribution amplitude of the $\eta_c$ meson, and to deal
with the evolution of the distribution amplitude to the effective energy scale
of the process. Our results show that one can obtain a compatible prediction
with the Belle and BarBar experimental data.
