Measurement of double charmonium production in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>e</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>e</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math>annihilations at<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msqrt><mml:mi>s</mml:mi></mml:msqrt><mml:mo>=</mml:mo><mml:mn>10.6</mml:mn><mml:mtext> </mml:mtext><mml:mtext> </mml:mtext><mml:mi>GeV</mml:mi></mml:math>

We study ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}J/\ensuremath{\psi}\text{ }\text{ }c\overline{c}$ by measuring the invariant mass distribution recoiling against fully reconstructed $J/\ensuremath{\psi}$ decays, using $124\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of data collected at a center-of-mass energy of 10.6 GeV with the BABAR detector. We observe signals for ${\ensuremath{\eta}}_{c}(1S)$, ${\ensuremath{\chi}}_{c0}$, and ${\ensuremath{\eta}}_{c}(2S)$ in the recoil mass distribution, thus confirming previous measurements. We measure $\ensuremath{\sigma}({e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}J/\ensuremath{\psi}+c\overline{c})\mathcal{B}(c\overline{c}\ensuremath{\rightarrow}&gt;2\text{ }\text{ }\mathrm{\text{charged}})$ to be $17.6\ifmmode\pm\else\textpm\fi{}2.8{(\mathrm{stat})}_{\ensuremath{-}2.1}^{+1.5}(\mathrm{syst})\text{ }\text{ }\mathrm{fb}$, $10.3\ifmmode\pm\else\textpm\fi{}2.5{(\mathrm{\text{stat}})}_{\ensuremath{-}1.8}^{+1.4}(\mathrm{\text{syst}})\text{ }\text{ }\mathrm{fb}$, and $16.4\ifmmode\pm\else\textpm\fi{}3.7{(\mathrm{\text{stat}})}_{\ensuremath{-}3.0}^{+2.4}(\mathrm{\text{syst}})\text{ }\text{ }\mathrm{fb}$ with $c\overline{c}={\ensuremath{\eta}}_{c}(1S)$, ${\ensuremath{\chi}}_{c0}$, and ${\ensuremath{\eta}}_{c}(2S)$, respectively.