We study the phase behavior of a nonionic ternary surfactant system to which we add small amounts of an ionic surfactant. At sufficiently high cosurfactant to surfactant ratio, we find at moderate concentration in amphiphiles a non-birefringent phase having very elastic rheological properties analogous to those previously reported by Hoffmann et al. (Langmuir 1994, 10, 3972) for similar systems. Our neutron scattering study agrees with the onion structure found by these authors for this remarkable phase. In addition, we also studied the structure of the fluid isotropic phase found at the same cosurfactant to surfactant ratios but in the dilute range. The present neutron scattering data indicate that it consists of a quasi-monodisperse dispersion of very small vesicles. We discuss the structures of these two phases, where the bilayers show finite curvatures, in terms of the bending elasticity of weakly charged membranes.