The first observation of the production of W^{±}W^{±} bosons from double parton scattering processes using same-sign electron-muon and dimuon events in proton-proton collisions is reported. The data sample corresponds to an integrated luminosity of 138 fb^{-1} recorded at a center-of-mass energy of 13 TeV using the CMS detector at the CERN LHC. Multivariate discriminants are used to distinguish the signal process from the main backgrounds. A binned maximum likelihood fit is performed to extract the signal cross section. The measured cross section for production of same-sign W bosons decaying leptonically is 80.7±11.2(stat) _{-8.6}^{+9.5}(syst)±12.1(model) fb, whereas the measured fiducial cross section is 6.28±0.81(stat)±0.69(syst)±0.37(model) fb. The observed significance of the signal is 6.2 standard deviations above the background-only hypothesis.