We study the effect of amphiphilic block copolymers (poly(oxyethylene)-poly(oxypropylene)-poly(oxyethylene)) on the phase behavior and elastic properties of a lyotropic lamellar phase (made from a mixture of cetylpiridinium chloride/octanol/water) into which they are incorporated. The hydrophobic blocks adsorb onto the bilayers, whereas the hydrophilic tails remain in the aqueous solvent and decorate them. Small-angle (neutron and X-ray) experiments were carried out to characterize the structural and physical properties of the mixed system. We observe that the host phase, that is, the lamellar phase, remains preserved over large ranges of polymer and surfactant concentrations. The effects of the polymer layers are 2-fold. On one hand, they modify the elastic properties of the individual membranes: (i) they soften the bilayer's mean curvature bending modulus and (ii) they stretch and therefore thin down the membranes; this thinning is due to a balance between the stretching of the polymer brush in the direction normal to the layers and the in-plane stretching of the bilayers. On the other hand, the presence of polymer layers grafted on the membranes strongly enhances the repulsive Helfrich interaction between the fluid membranes by increasing the effective bilayer thickness. This effect is strong, even at very low polymer concentration. The competition between the electrostatic and this "Helfrich's modified" interaction is also studied. A simple model is built up which accounts almost quantitatively for both the strong stiffening of the Bragg peaks and the gradual insensitiveness to added salt of charged bilayers decorated by increasing amounts of amphiphilic copolymers.