(W)e study a ternary mixture of oil, water, and surfactant in the case of equal volume fractions of oil and water using the Landau-Ginzburg model derived from a lattice model of this ternary mixture. We concentrate on a phase region close to a coexistence line between microemulsion and cubic phases. In our model the bicontinuous cubic phases exist in a narrow window of the volume fraction of surfactant rho(s)approximate to 0.6. The sequence of phase transitions is L-->G-->D-->P- ->C as we increaser rho(s) along the cubic-microemulsion bifurcation line. Here L stands for the lamellar phase and C for the cubic micella phase. The gyroid G, primitive P, and diamond D phases are all bicontinuous. The transitions weakly depend on the temperature. The increase of rho(s) is accompanied by the increase of the surface area per unit volume. In the case of fluctuating monolayers the interface is diffused and the average area of the monolayer per unit volume is larger than the "projected area,＇＇ i.e., the area of the surface describing the average position of the monolayer, per unit volume. The effect is the strongest in the L and the weakest in the C structure.