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This Letter reports a measurement of the exclusive γγ→ℓ+ℓ− (ℓ=e, μ) 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 View the MathML source for a phase–space region with invariant mass of the electron pairs greater than 24 GeV, in which both electrons have transverse momentum pT>12 GeV and pseudorapidity |η|<2.4. For muon pairs with invariant mass greater than 20 GeV, muon transverse momentum pT>10 GeV and pseudorapidity |η|<2.4, the cross-section is determined to be View the MathML source. 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.