We present results of a search for W^+W^- production through the
leptonic decay channel W^+W^- l^+l^- nu nu-bar in pbar p collisions at sqrt s
= 1.8 TeV.  In a 108 pb^-1 data sample recorded with the Collider Detector at
Fermilab, five W^+W^- candidates are found with an expected standard model
background of 1.2 +/- 0.3 events.  The W^+W^- production cross section is
measured to be sigma(pbar p -> W^+W^-) = 10.2 ^+6.3_ -5.1(stat) +\- 1.6(syst)
pb, in agreement with the standard model prediction.  Limits on WWgamma and
WWZ anomalous couplings are presented.