A Bose-Einstein condensate is the ground state of a dilute gas of bosons, such as atoms cooled to temperatures close to absolute zero(1). With much smaller mass, excitons (bound electron-hole pairs) are expected to condense at considerably higher temperatures(2-7). 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 10(12) excitons per square centimetre. The interlayer tunnelling current depends only on the exciton density, which is indicative of correlated electron-hole pair tunnelling(8). 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 temperature(9-12). 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 superconductivity(13).