The cross-section for the production of a single top quark in association with a W boson in proton-proton collisions at 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 W t signal is separated from the backgrounds using boosted decision trees, each of which combines a number of discriminating variables into one classifier. Production of W t events is observed with a significance of 7.7 sigma. The cross-section is extracted in a profile likelihood fit to the classifier output distributions. The W t cross-section, inclusive of decay modes, is measured to be 23.0 +/- 1.3(stat.) (-aEuro parts per thousand 3.5) (+ 3.2) (syst.)+/- 1.1(lumi.) pb. The measured cross-section is used to extract a value for the CKM matrix element |V (tb) | 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 p (T) > 25 GeV and |eta| < 2.5, one jet with p (T) > 20 GeV and |eta| < 2.5, and E (T) (miss) > 20 GeV, including both W t and top-quark pair events as signal. The measured value of the fiducial cross-section is 0.85 +/- A 0.01(stat.) (-aEuro parts per thousand 0.07) (+ 0.07) (syst.)+/- 0.03(lumi.) pb.