This Letter reports a measurement of the exclusive gamma gamma -> l(+)l(-) (l = e, mu) cross-section in proton-proton collisions at a centre-of-mass energy of 7 TeV by the ATLAS experiment at the LHC, based on an integrated luminosity of 4.6 fb(-1). For the electron or muon pairs satisfying exclusive selection criteria, a fit to the dilepton acoplanarity distribution is used to extract the fiducial cross-sections. The cross-section in the electron channel is determined to be sigma(excl)(gamma gamma -> e+e-) = 0.428 +/- 0.035 (stat.) +/- 0.018 (syst.) pbfor a phase-space region with invariant mass of the electron pairs greater than 24GeV, in which both electrons have transverse momentum p(T) > 12 GeV and pseudorapidity vertical bar eta vertical bar < 2.4. For muon pairs with invariant mass greater than 20GeV, muon transverse momentum pT> 10 GeV and pseudorapidity vertical bar eta vertical bar < 2.4, the cross-section is determined to be sigma(excl)(gamma gamma -> mu+mu-) = 0.628 +/- 0.032(stat.) +/- 0.021 (syst.) pb. When proton absorptive effects due to the finite size of the proton are taken into account in the theory calculation, the measured cross-sections are found to be consistent with the theory prediction. (C) 2015 CERN for the benefit of the ATLAS Collaboration. Published by Elsevier B.V.