We present an observation of photon-photon production of τ lepton pairs in ultraperipheral lead-lead collisions. The measurement is based on a data sample with an integrated luminosity of 404 μb(-1) collected by the CMS experiment at a center-of-mass energy per nucleon pair of √s(NN) = 5.02 TeV. The γγ → τ(+)τ(-) process is observed for τ(+)τ(-) events with a muon and three charged hadrons in the final state. The measured fiducial cross section is σ(γγ -→ τ(+)τ(-)) = 4.8 ± 0.6(stat) ± 0.5(syst) μb, where the second (third) term corresponds to the statistical (systematic) uncertainty in σ(γγ -→ τ(+)τ(-)) in agreement with leading-order QED predictions. Using σ(γγ -→ τ(+)τ(-)), we estimate a model-dependent value of the anomalous magnetic moment of the τ lepton of at a(τ) = 0.001(-0.089)(+0.055).