  This letter reports the observation of photon-induced production of $W$-boson
pairs, $\gamma\gamma\rightarrow WW$. The analysis uses 139 fb$^{-1}$ of LHC
proton-proton collision data taken at $\sqrt{s}=13$ 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
$WW\rightarrow e^\pm\nu\mu^\mp\nu$ 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
$\gamma\gamma\rightarrow WW$ process of 3.13$\pm$0.31(stat.)$\pm$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.
