The commonly held model for membrane dissolution by detergents/surfactants requires lipid transport from the inner to the outer bilayer leaflet ('flip-flop'). Although applicable to many systems, it fails in cases where cross-bilayer transport of membrane components is suppressed. In this paper we investigate the mechanism for surfactant-induced solubilization of polymeric bilayers. To that end, we examine the dissolution of a series of increasingly thick, polymer-based vesicles (polymersomes) by a nonionic surfactant, Triton X-100, using dynamic light scattering. We find that increasing the bilayer thickness imparts better resistance to dissolution, so that the concentration required for solubilization, after a fixed amount of time, increases nearly linearly with membrane thickness. Combining our experimental data with a theoretical model, we show that the dominant mechanism for the surfactant-induced dissolution of polymeric vesicles, where polymer flip-flop across the membrane is suppressed, is the surfactant transport through the bilayer. This mechanism is different both qualitatively and quantitatively from the mechanisms by which surfactants dissolve pure lipid vesicles.