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 significantly higher temperatures. Here we study electrically generated interlayer excitons in MoSe2/WSe2 atomic double layers with density up to 10^12 cm-2. The interlayer tunneling current depends only on exciton density, indicative of correlated electron-hole pair tunneling. Strong electroluminescence (EL) arises when a hole tunnels from WSe2 to recombine with electron in MoSe2. We observe a critical threshold dependence of the EL intensity on exciton density, accompanied by a super-Poissonian photon statistics near threshold, and a large EL enhancement peaked narrowly at equal electron-hole densities. The phenomenon persists above 100 K, which is consistent with the predicted critical condensation temperature. Our study provides compelling evidence for interlayer exciton condensation in two-dimensional atomic double layers and opens up exciting opportunities for exploring condensate-based optoelectronics and exciton-mediated high-temperature superconductivity.