We have investigated a section through the composition-temperature phase prism of a ternary nonionic surfactant-water-oil system defined by a constant surfactant to water ratio. The structural sequence normal cylinders-multiply connected bilayer-planar bilayer-reverse monolayer structure is obtained when increasing the temperature. This sequence corresponds to a gradual reversal of the mean curvature of the polar-apolar interface as the temperature is increased as has been observed previously in the corresponding microemulsion systems. The cylinders can incorporate (swell with) moderate amounts of oil. In addition to the bicontinuous cubic phase present in the binary water-surfactant system the ternary system contains a bicontinuous cubic phase with simultaneously high contents of water and oil. Upon dilution with oil, this phase undergoes an order-disorder transition to a liquid ''sponge phase' (L4) retaining the multiply connected bilayer structure. A lamellar phase swells with large amounts of oil. From the variation of the periodicity with oil content we obtain the solubility of the nonionic surfactant in the oil.