This paper presents a measurement of the W+W- production cross section in pp collisions at root s = 7 TeV. The leptonic decay channels are analyzed using data corresponding to an integrated luminosity of 4: 6 fb(-1) collected with the ATLAS detector at the Large Hadron Collider. The W+W- production cross section sigma(pp -> W+W- + X) is measured to be 51.9 +/- 2.0(stat) +/- 3.9(syst) +/- 2.0(lumi) pb, compatible with the Standard Model prediction of 44.7(-1.9)(+2.1) pb. A measurement of the normalized fiducial cross section as a function of the leading lepton transverse momentum is also presented. The reconstructed transverse momentum distribution of the leading lepton is used to extract limits on anomalous WWZ and WW gamma couplings.