This Letter reports a measurement of the WW production cross section in sqrt(s) = 7 TeV pp collisions using data corresponding to an integrated luminosity of 1.02/fb collected with the ATLAS detector. Using leptonic decays of oppositely charged W bosons, the total measured cross section is sigma(pp -> WW) = 54.4 +/- 4.0 (stat.) +/- 3.9 (syst.) +/- 2.0 (lumi.) pb, consistent with the Standard Model prediction of sigma(pp -> WW) = 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.