A Bose–Einstein condensate is the ground state of a dilute gas of bosons, such as atoms cooled to temperatures close to absolute zero1. With much smaller mass, excitons (bound electron–hole pairs) are expected to condense at considerably higher temperatures2–7. Two-dimensional van der Waals semiconductors with very strong exciton binding are ideal systems for the study of high-temperature exciton condensation. Here we study electrically generated interlayer excitons in MoSe2–WSe2 atomic double layers with a density of up to 1012 excitons per square centimetre. The interlayer tunnelling current depends only on the exciton density, which is indicative of correlated electron–hole pair tunnelling8. Strong electroluminescence arises when a hole tunnels from WSe2 to recombine with an electron in MoSe2. We observe a critical threshold dependence of the electroluminescence intensity on exciton density, accompanied by super-Poissonian photon statistics near the threshold, and a large electroluminescence enhancement with a narrow peak at equal electron and hole densities. The phenomenon persists above 100 kelvin, which is consistent with the predicted critical condensation temperature9–12. Our study provides evidence for interlayer exciton condensation in two-dimensional atomic double layers and opens up opportunities for exploring condensate-based optoelectronics and exciton-mediated high-temperature superconductivity13. © 2019, The Author(s), under exclusive licence to Springer Nature Limited.