A Bose-Einstein condensate is the ground state of a dilute gas of bosons, such as atoms cooled to temperatures close to absolute zero. With much smaller mass, excitons (bound electron-hole pairs) are expected to condense at considerably higher temperatures. 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 MoSe-WSe atomic double layers with a density of up to 10 excitons per square centimetre. The interlayer tunnelling current depends only on the exciton density, which is indicative of correlated electron-hole pair tunnelling. Strong electroluminescence arises when a hole tunnels from WSe to recombine with an electron in MoSe. 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 temperature. 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 superconductivity.