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 sqrt[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 a_{τ}=0.001_{-0.089}^{+0.055}.