A search for the charged-lepton-flavor-violating process $Z\ensuremath{\rightarrow}e\ensuremath{\mu}$ is presented, using $139\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ $pp$ collision data collected by the ATLAS experiment at the LHC. An excess in the $e\ensuremath{\mu}$ invariant mass spectrum near the $Z$ boson mass would be a striking signature of new physics. No excess is observed, and an upper limit $\mathcal{B}(Z\ensuremath{\rightarrow}e\ensuremath{\mu})<2.62\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ is placed on the branching fraction at 95% confidence level, which is the most stringent limit to date.