A measurement of the production of W bosons with opposite electric charges in association with two jets is presented based on 140 fb−1 of data collected by the ATLAS detector in proton-proton collisions at s = 13 TeV. The analysis is sensitive to the scattering of W bosons, which is of particular interest in the ATLAS physics programme as it can be used to probe the electroweak symmetry breaking mechanism of the Standard Model. This signal is observed with a significance of 7.1 standard deviations above the background expectation, while 6.2 standard deviations were expected. The measured cross-section is determined in a signal-enriched fiducial volume and is found to be 2.7 ± 0.5 fb, which is consistent with the theoretical prediction of 2.20−0.13+0.14 fb.