The binary phase diagram of water–C₁₂E₅ has been studied with emphasis on the L₃ and dilute lamellar phases, which were found to swell to approximately 99.5 and 98.8 wt % of water, respectively, much further than has been reported previously. Focusing on these two phases, we have carried out static light and small-angle neutron scattering and electrical conductivity measurements. The repeat distance in the lamellar phase was found to exceed 3000 Å. A small, but significant deviation from ideal one-dimensional swelling was observed. This deviation may be explained in terms of flexibility of the bilayers that are flat only on average. Electrical conductivity and small-angle neutron scattering data from the isotropic L₃ phase are, over most of the stability range, consistent with a three-dimensional continuous bilayer structure. However, at large water contents an increase in the conductivity indicates a breakup of the structure into smaller fragments. Our results show that the simpler binary system exhibits the same characteristic features as the more complex multicomponent systems, involving brine, ionic surfactant and cosurfactant.