We report the observation of Higgs boson decays to WW∗ based on an excess over background of 6.1 standard deviations in the dilepton final state, where the Standard Model expectation is 5.8 standard deviations. Evidence for the vector-boson fusion (VBF) production process is obtained with a significance of 3.2 standard deviations. The results are obtained from a data sample corresponding to an integrated luminosity of 25 fb−1 from √s=7 and 8 TeV pp collisions recorded by the ATLAS detector at the LHC. For a Higgs boson mass of 125.36 GeV, the ratio of the measured value to the expected value of the total production cross section times branching fraction is 1.09+0.16−0.15(stat)+0.17−0.14(syst). The corresponding ratios for the gluon fusion and vector-boson fusion production mechanisms are 1.02±0.19(stat)+0.22−0.18(syst) and 1.27+0.44−0.40(stat)+0.30−0.21(syst), respectively. At √s=8 TeV, the total production cross sections are measured to be σ(gg→H→WW∗)=4.6±0.9(stat)+0.8−0.7(syst) pb and σ(VBF H→WW∗)=0.51+0.17−0.15(stat)+0.13−0.08(syst) pb. The fiducial cross section is determined for the gluon-fusion process in exclusive final states with zero or one associated jet.