This Letter describes the current most precise measurement of the W boson pair production cross section and most sensitive test of anomalous WWγ and WWZ couplings in pp 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  fb−1 of integrated luminosity, a total of 654 candidate events are observed with an expected background of 320±47 events. The measured cross section is σ(pp→W+W−+X)=12.1±0.9(stat)+1.6−1.4(syst)  pb, which is in good agreement with the standard model prediction. The same data sample is used to place constraints on anomalous WWγ and WWZ couplings.