This letter reports the observation of photon-induced production of W-boson pairs, gamma gamma -> WW. The analysis uses 139 fb(-1) of LHC proton-proton collision data taken at root s = 13 TeV recorded by the ATLAS experiment during the years 2015-2018. The measurement is performed selecting one electron and one muon, corresponding to the decay of the diboson system as WW -> e(+/-)nu mu(-/+)nu final state. The background-only hypothesis is rejected with a significance of well above 5 standard deviations consistent with the expectation from Monte Carlo simulation. A cross section for the gamma gamma -> WW process of 3.13 +/- 0.31(stat.) +/- 0.28(syst.) fb is measured in a fiducial volume close to the acceptance of the detector, by requiring an electron and a muon of opposite signs with large dilepton transverse momentum and exactly zero additional charged particles. This is found to be in agreement with the Standard Model prediction. (C) 2021 The Author(s). Published by Elsevier B.V.