Double-β-decay involves the simultaneous conversion of two neutrons into two protons, and the emission of two electrons and two neutrinos; the neutrinoless process, although not yet observed, is thought to involve the emission of the two electrons but no neutrinos. The search for neutrinoless-double-β-decay probes fundamental properties of neutrinos, including whether or not the neutrino and antineutrino are distinct particles. Double-β-decay detectors are large and expensive, so it is essential to achieve the highest possible sensitivity with each study, and removing spurious contributions ('background') from detected signals is crucial. In the nEXO neutrinoless-double-β-decay experiment, the identification, or 'tagging', of the 136Ba daughter atom resulting from the double-β decay of 136Xe provides a technique for discriminating background. The tagging scheme studied here uses a cryogenic probe to trap the barium atom in a solid xenon matrix, where the barium atom is tagged through fluorescence imaging. Here we demonstrate the imaging and counting of individual barium atoms in solid xenon by scanning a focused laser across a solid xenon matrix deposited on a sapphire window. When the laser irradiates an individual atom, the fluorescence persists for about 30 seconds before dropping abruptly to the background level-a clear confirmation of one-atom imaging. Following evaporation of a barium deposit, the residual barium fluorescence is 0.16 per cent or less. Our technique achieves the imaging of single atoms in a solid noble element, establishing the basic principle of barium tagging for nEXO.