This Letter reports the observation of WWW production and a measurement of its cross section using 139  fb^{-1} of proton-proton collision data recorded at a center-of-mass energy of 13 TeV by the ATLAS detector at the Large Hadron Collider. Events with two same-sign leptons (electrons or muons) and at least two jets, as well as events with three charged leptons, are selected. A multivariate technique is then used to discriminate between signal and background events. Events from WWW production are observed with a significance of 8.0 standard deviations, where the expectation is 5.4 standard deviations. The inclusive WWW production cross section is measured to be 820±100 (stat)±80 (syst)  fb, approximately 2.6 standard deviations from the predicted cross section of 511±18  fb calculated at next-to-leading-order QCD and leading-order electroweak accuracy.