  In this work we evaluate the cross section of the process $e^+e^-\to J/\psi
\eta_c$ at energy $\sqrt{s}\approx 10.6$ GeV in the Bethe-Salpeter formalism.
To simplify our calculation, the heavy quark limit is employed. Without taking
the beyond-leading-order contribution(s) into account, the cross section
calculated in this scenario is comparable with the experimental data. We also
present our prediction for the cross section of double bottomonium production
$e^+e^-\to \Upsilon(1S)\eta_b$ for the energy range of $\sqrt{s}\approx (25
\hbox{-} 30)$ GeV which may be experimentally tested, even though there is no
facility of this range available at present yet.
