We present the binary phase diagram of the system dodecaoxyethylene mono-n-dodecyl ether (C-12-EO12)/water, which is the first pure surfactant system found to exhibit three different type I (oil-in-water) micellar cubic phases. As hydration increases the hexagonal, H-I, phase transforms into a cubic, I-1, phase of space group Pm3n, which, on further hydration, forms a second micellar cubic phase of space group Im3m (phase designated as Im3m). In addition, a third micellar cubic phase, of space group Fm3m, farms at low temperature and high hydration, adjacent to the L-1 micellar solution. We have succeeded in growing monodomains of the hexagonal, and some of these cubic phases and have thereby investigated the epitaxial relationships between the phases. The results suggest an "undulating cylinder" mechanism for the Im3m-H-I transition.