We report a measurement of the cross section of single top-quark production in the t-channel using 1.04 fb(-1) of pp collision data at root s = 7 TeV recorded with the ATLAS detector at the LHC. Selected events feature one electron or muon, missing transverse momentum, and two or three jets, exactly one of them identified as originating from a b quark. The cross section is measured by fitting the distribution of a multivariate discriminant constructed with a neural network, yielding sigma(t) = 83 +/- 4 (stat.)(-19)(+20) (syst.) pb, which is in good agreement with the prediction of the Standard Model. Using the ratio of the measured to the theoretically predicted cross section and assuming that the top-quark-related CKM matrix elements obey the relation vertical bar V-tb vertical bar >> vertical bar V-ts vertical bar, vertical bar V-td vertical bar, the coupling strength at the W-t-b vertex is determined to be vertical bar V-tb vertical bar = 1.13(-0.13)(+0.14). If it is assumed that vertical bar V-tb vertical bar <= 1 a lower limit of vertical bar V-tb vertical bar > 0.75 is obtained at the 95% confidence level.