Microporous polymers of extreme rigidity are required for gas-separation membranes that combine high permeability with selectivity. We report a shape-persistent ladder polymer consisting of benzene rings fused together by inflexible bridged bicyclic units. The polymer's contorted shape ensures both microporosity-with an internal surface area greater than 1000 square meters per gram-and solubility so that it is readily cast from solution into robust films. These films demonstrate exceptional performance as molecular sieves with high gas permeabilities and good selectivities for smaller gas molecules, such as hydrogen and oxygen, over larger molecules, such as nitrogen and methane. Hence, this polymer has excellent potential for making membranes suitable for large-scale gas separations of commercial and environmental relevance.