The observation of the electroweak production of single-top-quarks is made using 255 pb^-1 of proton-proton collision data recorded at √s = 5.02 TeV with the ATLAS detector at the Large Hadron Collider. An event selection is used to identify single-top-quark candidates arising from..-channel production with the top quark decaying semi-leptonically. Events passing the selection are then used to measure the inclusive cross-section for the combined production of single-top-quarks and antiquarks, σ(tq + t<over bar>q), and the ratio R_t between these two. They are measured to be σ(tq + t<over bar>q = 27.1_-4.1^+4.4(stat.)_-3.7^+4.4(syst.) pb and R_t = 2.73_-0.82^+1.43(stat.)_-0.29^+1.01)(syst.). The individual single-top-quark (tq) and single-top-antiquark (t<over bar>q) production cross-sections are measured to be σ(tq) = 19.8_-3.1^+3.9(stat.)_-2.2^+2.9(syst.) pb and σ(t<over bar>q) = 7.3_-2.1^+3.2)(stat.)_-1.5^+2.8)(syst.) pb. All measurements are in good agreement with the Standard Model predictions.