Effects of confinement on the cubic bicontinuous phase with double-diamond symmetry (D phase) are studied in the lattice, Ciach-Hoye-Stell model of oil-water-surfactant mixtures within mean-field approximation. We focus on the system confined between two parallel hydrophilic walls, which are perpendicular to the diagonal of the cubic cell. The solvation force decays very slowly, and oscillates with the period lambda/3, where lambda is the edge length of the conventional cubic unit cell. The mechanical properties depend on the spectrum of the bulk metastable phases, on the difference between surfactant concentrations in the coexisting D and lamellar phases, Deltarho(s), and on the thermodynamic state. The smaller Deltarho(s) and the closer to the transition to the lamellar phase, the stronger the deviations from the elastic response to compression or expansion. The inelastic behavior is associated with formation of lamellar wetting films at the surfaces, and the stress is accommodated in the near-surface regions, whereas the D phase in the central part of the slit is only weakly deformed. The analog of capillary condensation of the lamellar phase does not obey the Kelvin equation, due to periodic nature and long range of the solvation force in both phases. When the wall separation L is decreased for a fixed thermodynamic state, a sequence of transitions between the D and lamellar phases occurs for L spanning decades of lambda. (C) 2003 American Institute of Physics.