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 root s = 7 TeV. Production via top decay is not included in the signal definition. The measurement is based on 35 pb(-1) of data collected with the ATLAS detector at the LHC. The W + b-jet cross section is defined for jets reconstructed with the anti-k(t) 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) 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.