At the critical micelle concentration (CMC), amphiphiles self-assemble into spherical micelles, typically followed by a transition at the second CMC to cylindrical micelles that are uniform in width but are polydispersed in length and have swollen ends. In this Letter, we report on a new structural path of self-assembly that is based on discoidal (coin-like), rather than spherical, geometry; the nonionic sterol ChEO10 is shown to form monodisperse equilibrium disc assemblies at the first CMC, transitioning at the second CMC into flat ribbons that (like the cylindrical micelles) have uniform width, polydispersed length, and swollen ends. Increase in ChEO10 concentration or the temperature leads to ribbon elongation, branching, and network formation. This self-assembly path reveals that (1) surfactants can form equilibrium nonspherical assemblies at the CMC and (2) aggregate progression around the second CMC is similar for the disc and sphere geometries.