We present a measurement of the W boson pair-production cross section in p collisions at a center-of-mass energy of =1.96 TeV. The data, collected with the Run II D0 detector at Fermilab, correspond to an integrated luminosity of 224–252 pb-1 depending on the final state (ee, eµ, or µµ). We observe 25 candidates with a background expectation of 8.1±0.6(stat)±0.6(syst)±0.5(lum) events. The probability for an upward fluctuation of the background to produce the observed signal is 2.3×10-7, equivalent to 5.2 standard deviations. The measurement yields a cross section of 13.8(stat)(syst)±0.9(lum) pb, in agreement with predictions from the standard model.