We present a measurement of the W+W− production cross section using 184  pb−1 of pp collisions at a center-of-mass energy of 1.96 TeV collected with the Collider Detector at Fermilab. Using the dilepton decay channel W+W−→ℓ+νℓ−ν¯, where the charged leptons can be either electrons or muons, we find 17 candidate events compared to an expected background of 5.0+2.2−0.8 events. The resulting W+W− production cross-section measurement of σ(pp→W+W−)=14.6+5.8−5.1(stat)+1.8−3.0(syst)±0.9(lum)  pb agrees well with the standard model expectation.