  This Letter reports a measurement of the exclusive $\gamma\gamma\rightarrow
\ell^+\ell^- (\ell=e, \mu)$ 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 $\sigma_{\gamma\gamma\rightarrow
e^+e^-}^{\mathrm{excl.}} = 0.428 \pm 0.035 (\mathrm{stat.}) \pm 0.018
(\mathrm{syst.})$ pb for a phase-space region with invariant mass of the
electron pairs greater than 24 GeV, in which both electrons have transverse
momentum $p_\mathrm{T}>12$ GeV and pseudorapidity $|\eta|<2.4$. For muon pairs
with invariant mass greater than 20 GeV, muon transverse momentum
$p_\mathrm{T}>10$ GeV and pseudorapidity $|\eta|<2.4$, the cross-section is
determined to be $\sigma_{\gamma\gamma\rightarrow \mu^+\mu^- }^{\mathrm{excl.}}
= 0.628 \pm 0.032 (\mathrm{stat.}) \pm 0.021 (\mathrm{syst.})$ 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.
