We present a search for the Higgs boson in final states with two oppositely charged leptons and large missing transverse energy as expected in $H\ensuremath{\rightarrow}WW\ensuremath{\rightarrow}\ensuremath{\ell}\ensuremath{\nu}{\ensuremath{\ell}}^{\ensuremath{'}}{\ensuremath{\nu}}^{\ensuremath{'}}$ decays. The events are selected from the full Run II data sample of $9.7\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $p\overline{p}$ collisions collected with the D0 detector at the Fermilab Tevatron Collider at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$. To validate our search methodology, we measure the nonresonant $WW$ production cross section and find ${\ensuremath{\sigma}}_{WW}=11.6\ifmmode\pm\else\textpm\fi{}0.7\text{ }\text{ }\mathrm{pb}$, in agreement with the standard model prediction. In the Higgs boson search, no significant excess above the background expectation is observed. Upper limits at the 95% confidence level on the Higgs boson production cross section are therefore derived. Within the standard model, the Higgs boson mass range $159<{M}_{H}<176\text{ }\text{ }\mathrm{GeV}$ is excluded while the expected exclusion sensitivity is $156<{M}_{H}<172\text{ }\text{ }\mathrm{GeV}$. For a mass hypothesis of ${M}_{H}=125\text{ }\text{ }\mathrm{GeV}$, we exclude Higgs boson production cross sections 4.1 times larger than the standard model expectation, which is compatible with the presence of a Higgs boson at this mass. Within a theoretical framework with a fourth generation of fermions, the mass range $125<{M}_{H}<218\text{ }\text{ }\mathrm{GeV}$ is excluded. The search results are also interpreted in the context of fermiophobic Higgs boson couplings, which yields an exclusion of fermiophobic Higgs boson production cross sections 3.1 times larger than the expectation for ${M}_{H}=125\text{ }\text{ }\mathrm{GeV}$.