Search for the process <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:mi>η</mml:mi></mml:mrow></mml:math>

A search for the rare decay $\ensuremath{\eta}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}$ is performed using the inverse process ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\eta}$ in the decay mode $\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$. We analyze data with an integrated luminosity of $654\text{ }\text{ }{\mathrm{nb}}^{\ensuremath{-}1}$ accumulated at the VEPP-2000 ${e}^{+}{e}^{\ensuremath{-}}$ collider with the SND detector at the center-of-mass energy $E={m}_{\ensuremath{\eta}}{c}^{2}\ensuremath{\approx}548\text{ }\text{ }\mathrm{MeV}$, and set the upper limit $\mathcal{B}(\ensuremath{\eta}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}})&lt;7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ at the 90% confidence level.