A search for the charged-lepton-flavor-violating process Z→eμ is presented, using 139  fb⁻¹ of √s=13  TeV pp collision data collected by the ATLAS experiment at the LHC. An excess in the eμ invariant mass spectrum near the Z boson mass would be a striking signature of new physics. No excess is observed, and an upper limit <2.62×10⁻⁷ is placed on the branching fraction at 95% confidence level, which is the most stringent limit to date.