This letter reports the measurement of double parton scattering in same-sign W boson pair production with the ATLAS detector. The data set used corresponds to an integrated luminosity of 140 fb−1 of proton–proton collisions at a center-of-mass energy of 13 TeV, collected during Run 2 of the Large Hadron Collider. The study is performed in final states including two same-charge leptons, electron or muon, missing transverse momentum, and up to one jet. An excess of events is observed over the expected background contributions with a significance of 8.8 standard deviations. The measured fiducial cross section times leptonic branching fraction is 4.59±0.64 fb. The measurement corresponds to a double parton scattering effective cross section of 10.6±1.8 mb.