A search for the bb¯ decay of the Standard Model Higgs boson is performed with the ATLAS experiment using the full dataset recorded at the LHC in Run 1. The integrated luminosities used are 4.7 and 20.3 fb−1 from pp collisions at √s=7 and 8 TeV, respectively. The processes considered are associated (W/Z)H production, where W → eν/μν, Z → ee/μμ and Z → νν. The observed (expected) deviation from the background-only hypothesis corresponds to a significance of 1.4 (2.6) standard deviations and the ratio of the measured signal yield to the Standard Model expectation is found to be μ = 0.52 ± 0.32 (stat.) ± 0.24 (syst.) for a Higgs boson mass of 125.36 GeV. The analysis procedure is validated by a measurement of the yield of (W/Z)Z production with Z→bb¯ in the same final states as for the Higgs boson search, from which the ratio of the observed signal yield to the Standard Model expectation is found to be 0.74 ± 0.09 (stat.) ± 0.14 (syst.).