A measurement of W+W- production in pp collisions at sqrt(s) = 7 TeV is presented. The data were collected with the CMS detector at the LHC, and correspond to an integrated luminosity of 4.92 +/- 0.11 inverse femtobarns. The W+W- candidates consist of two oppositely charged leptons, electrons or muons, accompanied by large missing transverse energy. The W+W- production cross section is measured to be 52.4 +/- 2.0 (stat.) +/- 4.5 (syst.) +/- 1.2 (lum.) pb. This measurement is consistent with the standard model prediction of 47.0 +/- 2.0 pb at next-to-leading order. Stringent limits on the WW gamma and WWZ anomalous triple gauge-boson couplings are set.