This Letter describes the current most precise measurement of the $W$ boson pair production cross section and most sensitive test of anomalous $WW\ensuremath{\gamma}$ and $WWZ$ couplings in $p\overline{p}$ collisions at a center-of-mass energy of 1.96 TeV. The $WW$ candidates are reconstructed from decays containing two charged leptons and two neutrinos. Using data collected by the CDF II detector from $3.6\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of integrated luminosity, a total of 654 candidate events are observed with an expected background of $320\ifmmode\pm\else\textpm\fi{}47$ events. The measured cross section is $\ensuremath{\sigma}(p\overline{p}\ensuremath{\rightarrow}{W}^{+}{W}^{\ensuremath{-}}+X)=12.1\ifmmode\pm\else\textpm\fi{}0.9(\mathrm{stat}{)}_{\ensuremath{-}1.4}^{+1.6}(\mathrm{syst})\text{ }\text{ }\mathrm{pb}$, which is in good agreement with the standard model prediction. The same data sample is used to place constraints on anomalous $WW\ensuremath{\gamma}$ and $WWZ$ couplings.