We report on the results of a search for standard model Higgs bosons produced in association with W bosons from pp collisions at s√=1.96  TeV. The search uses a data sample corresponding to approximately 1  fb−1 of integrated luminosity. Events consistent with the W→ℓν and H→bb signature are selected by triggering on a high-pT electron or muon candidate and tagging one or two of the jet candidates as having originated from b quarks. A neural network filter rejects a fraction of tagged charm and light-flavor jets, increasing the b-jet purity in the sample. We observe no excess ℓνbb production beyond the background expectation, and we set 95% confidence level upper limits on the production cross section times branching fraction σ(pp→WH)Br(H→bb) ranging from 3.9 to 1.3 pb, for specific Higgs boson mass hypotheses in the range 110 to 150  GeV/c2, respectively.