Pore size uniformity is one of the most critical parameters in determining membrane separation performance. Recently, a novel type of conjugated microporous polymers (CMPs) has shown uniform pore size and high porosity. However, their brittle nature has prevented them from preparing robust membranes. Inspired by the skin-core architecture of spider silk that offers both high strength and high ductility, herein we report an electropolymerization process to prepare a CMP membrane from a rigid carbazole monomer, 2,2',7,7'-tetra(carbazol-9-yl)-9,9'-spirobifluorene, inside a robust carbon nanotube scaffold. The obtained membranes showed superior mechanical strength and ductility, high surface area, and uniform pore size of approximately 1 nm. The superfast solvent transport and excellent molecular sieving well surpass the performance of most reported polymer membranes. Our method makes it possible to use rigid CMPs membranes in pressure-driven membrane processes, providing potential applications for this important category of polymer materials.