Major galaxy mergers are thought to play an important part in fuelling the growth of supermassive black holes1. However, observational support for this hypothesis is mixed, with some studies showing a correlation between merging galaxies and luminous quasars2,3 and others showing no such association4,5. Recent observations have shown that a black hole is likely to become heavily obscured behind merger-driven gas and dust, even in the early stages of the merger, when the galaxies are well separated6-8 (5 to 40 kiloparsecs). Merger simulations further suggest that such obscuration and black-hole accretion peaks in the final merger stage, when the two galactic nuclei are closely separated9 (less than 3 kiloparsecs). Resolving this final stage requires a combination of high-spatial-resolution infrared imaging and high-sensitivity hard-X-ray observations to detect highly obscured sources. However, large numbers of obscured luminous accreting supermassive black holes have been recently detected nearby (distances below 250 megaparsecs) in X-ray observations10. Here we report high-resolution infrared observations of hard-X-ray-selected black holes and the discovery of obscured nuclear mergers, the parent populations of supermassive-black-hole mergers. We find that obscured luminous black holes (bolometric luminosity higher than 2 × 1044 ergs per second) show a significant (P < 0.001) excess of late-stage nuclear mergers (17.6 per cent) compared to a sample of inactive galaxies with matching stellar masses and star formation rates (1.1 per cent), in agreement with theoretical predictions. Using hydrodynamic simulations, we confirm that the excess of nuclear mergers is indeed strongest for gas-rich major-merger hosts of obscured luminous black holes in this final stage.