When a vesicle, delimited by a flexible surfactant membrane, is confined to a slit or a cylindrical capillary, its accessible free volume is reduced. It is shown that the corresponding increase in free energy, taken as the sum of curvature elastic energy and of the lowered entropic contribution, leads to a change of the equilibrium shape, and to a significant shift of the prolate to oblate transition observed for an isolated vesicle, when the reduced bending rigidity is sufficiently small.