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Carbon molecular sieve membranes (CMSMs) were prepared by carbonizing the high free volume polyimide BTDA-BAF that is obtained from the reaction of benzophenone-3,3′,4,4′-tetracarboxylic dianhydride (BTDA) and 9,9-bis(4-aminophenyl) fluorene (BAF). The bulky cardo groups prevented a tight packing and rotation of the chains that leads to high permeabilities of their CMSMs. The incorporation of metal–organic polyhedra 18 (MOP-18, a copper-based MOP) in the BTDA-BAF polymer before pyrolysis at 550 °C prevented the collapse of the pores and the aging of the CMSMs. It was found that upon decomposition of MOP-18, a distribution of copper nanoparticles minimized the collapse of the graphitic sheets that formed the micropores and mesopores in the CMSM. The pillared CMSMs displayed CO2 and CH4 permeabilities of 12,729 and 659 Barrer, respectively, with a CO2/CH4 selectivity of 19.3 after 3 weeks of aging. The permselectivity properties of these membranes was determined to be at the 2019 Robeson upper bound. In contrast, the CMSMs from pure BTDA-BAF aged three times faster than the CMSMs from MOP-18/BTDA-BAF and exhibited lower CO2 and CH4 permeabilities of 5337 and 573 Barrer, respectively, with a CO2/CH4 selectivity of 9.3. The non-pillared CMSMs performed below the upper bound.