We present a measurement of the $W$ boson pair-production cross section in $p\overline{p}$ collisions at a center-of-mass energy of $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$. The data, collected with the Run II D0 detector at Fermilab, correspond to an integrated luminosity of $224--252\text{ }\text{ }{\mathrm{pb}}^{\ensuremath{-}1}$ depending on the final state ($ee$, $e\ensuremath{\mu}$, or $\ensuremath{\mu}\ensuremath{\mu}$). We observe 25 candidates with a background expectation of $8.1\ifmmode\pm\else\textpm\fi{}0.6(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.6(\mathrm{syst})\ifmmode\pm\else\textpm\fi{}0.5(\mathrm{lum})$ events. The probability for an upward fluctuation of the background to produce the observed signal is $2.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$, equivalent to 5.2 standard deviations. The measurement yields a cross section of ${13.8}_{\ensuremath{-}3.8}^{+4.3}(\mathrm{stat}{)}_{\ensuremath{-}0.9}^{+1.2}(\mathrm{syst})\ifmmode\pm\else\textpm\fi{}0.9(\mathrm{lum})\text{ }\text{ }\mathrm{pb}$, in agreement with predictions from the standard model.