  Using a data set of 2.93 fb$^{-1}$ taken at a center-of-mass energy of
$\sqrt{s}$ = 3.773 GeV with the BESIII detector at the BEPCII collider, we
measure the process $e^+e^-\rightarrow J/\psi\gamma\rightarrow
\mu^+\mu^-\gamma$ and determine the product of the branching fraction and the
electronic width $\mathcal B_{\mu\mu}\cdot \Gamma_{ee} = (333.4 \pm 2.5_{\rm
stat} \pm 4.4_{\rm sys})$~eV. Using the earlier-published BESIII result for
$\mathcal B_{\mu\mu}$ = (5.973 $\pm$ 0.007$_{\rm stat}$ $\pm$ 0.037$_{\rm
sys}$)\%, we derive the $J/\psi$ electronic width $\Gamma_{ee}$~= (5.58 $\pm$
0.05$_{\rm stat}$ $\pm$ 0.08$_{\rm sys}$) keV.
