We present new results for the current as a function of transmission rate in the one-dimensional totally asymmetric simple exclusion process (TASEP) with a blockage that lowers the jump rate at one site from one to r < 1. Exact finite- volume results serve to bound the allowed values for the current in the infinite system. This proves the existence of a nonequilibrium "phase transition," corresponding to an "immiscibility" gap in the allowed values of the asymptotic densities which the infinite system can have in a stationary state.