A search for diphoton resonances in the mass range between 10 and 70 GeV with the ATLAS experiment at the Large Hadron Collider (LHC) is presented. The analysis is based on pp collision data corresponding to an integrated luminosity of 138 fb$^{−1}$ at a centre-of-mass energy of 13 TeV recorded from 2015 to 2018. Previous searches for diphoton resonances at the LHC have explored masses down to 65 GeV, finding no evidence of new particles. This search exploits the particular kinematics of events with pairs of closely spaced photons reconstructed in the detector, allowing examination of invariant masses down to 10 GeV. The presented strategy covers a region previously unexplored at hadron colliders because of the experimental challenges of recording low-energy photons and estimating the backgrounds. No significant excess is observed and the reported limits provide the strongest bound on promptly decaying axion-like particles coupling to gluons and photons for masses between 10 and 70 GeV.[graphic not available: see fulltext]