This Letter reports the observation of τ-lepton-pair production in ultraperipheral lead-lead collisions Pb+Pb→Pb(γγ→ττ)Pb and constraints on the τ-lepton anomalous magnetic moment a_{τ}. The dataset corresponds to an integrated luminosity of 1.44 nb^{-1} of LHC Pb+Pb collisions at sqrt[s_{NN}]=5.02 TeV recorded by the ATLAS experiment in 2018. Selected events contain one muon from a τ-lepton decay, an electron or charged-particle track(s) from the other τ-lepton decay, little …