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 root 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 root s = 8 TeV, the total production cross sections are measured to be sigma(gg -> H -> WW*) = 4.6 +/- 0.9(stat)(-0.7)(+0.8)(syst) pb and sigma(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 jet.