Layered metal-organic frameworks would be a diverse source of crystalline sheets with nanometer thickness for molecular sieving if they could be exfoliated, but there is a challenge in retaining the morphological and structural integrity. We report the preparation of 1-nanometer-thick sheets with large lateral area and high crystallinity from layered MOFs. They are used as building blocks for ultrathin molecular sieve membranes, which achieve hydrogen gas (H-2) permeance of up to several thousand gas permeation units (GPUs) with H-2/CO2 selectivity greater than 200. We found an unusual proportional relationship between H-2 permeance and H-2 selectivity for the membranes, and achieved a simultaneous increase in both permeance and selectivity by suppressing lamellar stacking of the nanosheets.