The ratio of production cross sections of the W and Z bosons with exactly one associated jet is presented as a function of jet transverse momentum threshold. The measurement has been designed to maximise cancellation of experimental and theoretical uncertainties, and is reported both within a particle-level kinematic range corresponding to the detector acceptance and as a total cross-section ratio. Results are obtained with the ATLAS detector at the LHC in pp collisions at a centre-of-mass energy of 7 TeV using an integrated luminosity of 33 pb^-1. The results are compared with perturbative leading-order, leading-log, and next-to-leading-order QCD predictions, and are found to agree within experimental and theoretical uncertainties. The ratio is measured for events with a single jet with p_T > 30 GeV to be 8.73 +/- 0.30 (stat) +/- 0.40 (syst) in the electron channel, and $ 8.49 +/- 0.23 (stat) +/- 0.33 (syst) in the muon channel.