A measurement is presented of the cross section for the production of a W boson with one or two jets, of which at least one must be a b -jet, in pp collisions at √s=7 TeV. Production via top decay is not included in the signal definition. The measurement is based on 35 pb−135 pb−1 of data collected with the ATLAS detector at the LHC. The W+bW+b-jet cross section is defined for jets reconstructed with the anti-ktkt clustering algorithm with transverse momentum above 25 GeV and rapidity within ±2.1. The b-jets are identified by reconstructing secondary vertices. The fiducial cross section is measured both for the electron and muon decay channel of the W boson and is found to be 10.2±1.9(stat)±2.6(syst) pb10.2±1.9(stat)±2.6(syst) pb for one lepton flavour. The results are compared with next-to-leading order QCD calculations, which predict a cross section smaller than, though consistent with, the measured value.