The cross-section for the production of a single top quark in association with a W boson in proton-proton collisions at √s = 8 TeV is measured. The dataset corresponds to an integrated luminosity of 20.3 fb-1, collected by the ATLAS detector in 2012 at the Large Hadron Collider at CERN. Events containing two leptons and one central b-jet are selected. The Wt signal is separated from the backgrounds using boosted decision trees, each of which combines a number of discriminating variables into one classifier. Production of Wt events is observed with a significance of 7.7σ. The cross-section is extracted in a profile likelihood fit to the classifier output distributions. The Wt cross-section, inclusive of decay modes, is measured to be 23.0±1.3 (stat.)+3.2 -3.5 (syst.) ±1.1(lumi.) pb. The measured cross-section is used to extract a value for the CKM matrix element |Vtb| of 1.01±0.10 and a lower limit of 0.80 at the 95% confidence level. The cross-section for the production of a top quark and a W boson is also measured in a fiducial acceptance requiring two leptons with pT > 25 GeV and |η| < 2:5, one jet with pT > 20 GeV and |η| < 2:5, and Emiss T > 20 GeV, including both Wt and top-quark pair events as signal. The measured value of the fiducial cross-section is 0.85±0.01(stat.)+0.06 -0.07(syst.)± 0.03(lumi.) pb.