This Letter reports a measurement of the W W production cross section in √s = 7 TeV pp collisions using data corresponding to an integrated luminosity of 1.02 fb−1 collected with the ATLAS detector. Using leptonic decays of oppositely charged W bosons, the total measured cross section is σ(pp → W W ) = 54.4 ± 4.0 (stat.) ± 3.9 (syst.) ± 2.0 (lumi.) pb, consistent with the Standard Model prediction of σ(pp → W W ) = 44.4 ± 2.8 pb. Limits on anomalous electroweak triple-gauge couplings are extracted from a fit to the transverse-momentum distribution of the leading charged lepton in the event.