The production of pure deuterium and the removal of tritium from nuclear waste are the key challenges in separation of light isotopes. Presently, the technological methods are extremely energy-and cost-intensive. Here we report the capture of heavy hydrogen isotopes from hydrogen gas by selective adsorption at Cu(I) sites in a metal-organic framework. At the strongly binding Cu(I) sites (32 kJ mol^-1) nuclear quantum effects result in higher adsorption enthalpies of heavier isotopes. The capture mechanism takes place most efficiently at temperatures above 80 K, when an isotope exchange allows the preferential adsorption of heavy isotopologues from the gas phase. Large difference in adsorption enthalpy of 2.5 kJ mol^-1 between D₂ and H₂ results in D₂-over-H₂ selectivity of 11 at 100 K, to the best of our knowledge the largest value known to date. Combination of thermal desorption spectroscopy, Raman measurements, inelastic neutron scattering and first principles calculations for H₂/D₂ mixtures allows the prediction of selectivities for tritium-containing isotopologues.