This Letter reports the observation of tau-lepton-pair production in ultraperipheral lead-lead collisions Pb + Pb -> Pb(gamma gamma -> tau tau)Pb and constraints on the tau-lepton anomalous magnetic moment a(tau). The dataset corresponds to an integrated luminosity of 1.44 nb(-1) of LHC Pb + Pb collisions at root(NN)-N-s = 5.02 TeV recorded by the ATLAS experiment in 2018. Selected events contain one muon from a t-lepton decay, an electron or charged-particle track(s) from the other tau-lepton decay, little additional central-detector activity, and no forward neutrons. The gamma gamma -> tau tau process is observed in Pb + Pb collisions with a significance exceeding 5 standard deviations and a signal strength of mu(tau tau) = 1.03(-0.05)(+0.06) assuming the standard model value for a(tau). To measure a(tau), a template fit to the muon transverse-momentum distribution from tau-lepton candidates is performed, using a dimuon (gamma gamma -> mu mu) control sample to constrain systematic uncertainties. The observed 95% confidence-level interval for a(tau) is -0.057 < a(tau) < 0.024.