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 1.09-0.15+0.16(stat)-0.14+0.17(syst). The corresponding ratios for the gluon fusion and vector-boson fusion production mechanisms are 1.02±0.19(stat)-0.18+0.22(syst) and 1.27-0.40+0.44(stat)-0.21+0.30(syst), respectively. At s=8 TeV, the total production cross sections are measured to be σ(gg→H→WW*)=4.6±0.9(stat)-0.7+0.8(syst) pb and σ(VBF H→WW*)=0.51-0.15+0.17(stat)-0.08+0.13(syst) pb. The fiducial cross section is determined for the gluon-fusion process in exclusive final states with zero or one associated