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