We present a measurement of the W(+)W(-) production cross section using 184 pb(-1) of p(p) 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(+)nul(-), 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 sigma(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.