CERN-LHC. 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 $7~\mathrm{TeV}$ by the ATLAS experiment at the LHC, based on an integrated luminosity of $4.6~\mathrm{fb}^{-1}$. 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 $\sigma_{\gamma\gamma\rightarrow e^+e^-}^{\mathrm{excl.}} = 0.428~\pm~0.035~(\mathrm{stat.})~\pm~0.018~(\mathrm{syst.})~\mathrm{pb}$ for a phase-space region with invariant mass of the electron pairs greater than $24~\mathrm{GeV}$, in which both electrons have transverse momentum $p_\mathrm{T}>12~\mathrm{GeV}$ and pseudorapidity $|\eta|<2.4$. For muon pairs with invariant mass greater than $20~\mathrm{GeV}$, muons transverse momentum $p_\mathrm{T}>10~\mathrm{GeV}$ and pseudorapidity $|\eta|<2.4$, the cross-section is determined to be $\sigma_{\gamma\gamma\rightarrow \mu^+\mu^- }^{\mathrm{excl.}} = 0.628~\pm~0.032~(\mathrm{stat.}) ~\pm~0.021~(\mathrm{syst.})~\mathrm{pb}$. When proton absorptive effects due to finite proton size are taken into account in the theory calculation, the measured cross-sections are found to be consistent with the prediction.