The exclusive production of pion pairs in the process $pp\rightarrow pp\pi ^+\pi ^-$ has been measured at $\sqrt{s}={7}\,\text {TeV}$ with the ATLAS detector at the LHC, using ${80}\,{\upmu \textrm{b}}^{-1}$ of low-luminosity data. The pion pairs were detected in the ATLAS central detector while outgoing protons were measured in the forward ATLAS ALFA detector system. This represents the first use of proton tagging to measure an exclusive hadronic final state at the LHC. A cross-section measurement is performed in two kinematic regions defined by the proton momenta, the pion rapidities and transverse momenta, and the pion–pion invariant mass. Cross-section values of $4.8 \pm 1.0 \mathrm {\ (stat)} {~}^{+0.3}_{-0.2} \mathrm {\ (syst)}\ {\upmu \textrm{b}}$ and $9 \pm 6 \mathrm {\ (stat)} {~}^{+2}_{-2} \mathrm {\ (syst)}\ {\upmu \textrm{b}} $ are obtained in the two regions; they are compared with theoretical models and provide a demonstration of the feasibility of measurements of this type.