This letter presents evidence for the associated production of a W boson and a top quark using 2.05 fb^-1 of pp collision data at sqrt(s) = 7 TeV accumulated with the ATLAS detector at the LHC. The analysis is based on the selection of the dileptonic final states with events featuring two isolated leptons, electron or muon, with significant transverse missing momentum and at least one jet. An approach based on boosted decision trees has been developed to improve the discrimination of single top-quark Wt events from background. A template fit to the final classifier distributions is performed to determine the cross-section. The result is incompatible with the background-only hypothesis at the 3.3 sigma level, the expected sensitivity assuming the Standard Model production rate being 3.4 sigma. The corresponding cross-section is determined and found to be sigma_Wt = 16.8 +/- 2.9 (stat) +/- 4.9 (syst) pb, in good agreement with the Standard Model expectation. From this result the CKM matrix element |V_{tb}| = 1.03{+0.16}{-0.19} is derived assuming that the Wt production through |V_{ts}| and |V_{td}| is small.