We study the phase behavior of ternary amphiphilic systems in the framework of a curvature model with nonvanishing spontaneous curvature. The amphiphilic monolayers can arrange in different ways to form micellar, hexagonal, lamellar, and various bicontinuous cubic phases. For the latter case we consider both single structures (one monolayer) and double structures (two monolayers). Their interfaces are modeled by the triply periodic surfaces of constant mean curvature of the families G, D, P, C(P), I-WP, and F-RD. The stability of the different bicontinuous cubic phases can be explained by the way in which their universal geometrical properties conspire with the concentration constraints. For vanishing saddle-splay modulus <(kappa)over bar>, almost every phase considered has some region of stability in the Gibbs triangle. Although bicontinuous cubic phases are suppressed by sufficiently negative values of the saddle-splay modulus <(kappa)over bar>, we find that they can exist for considerably lower values than obtained previously. The most stable bicontinuous cubic phases with decreasing <(kappa)over bar>< 0 are the single and double gyroid structures since they combine favorable topological properties with extreme volume fractions. (C) 2000 American Institute of Physics. [S0021-9606(00)70306-0].