We have investigated a lamellar phase (L-alpha) composed of a nonionic surfactant tetraethylene glycol monododecyl ether (C12E4), water, and decane, where a fraction of the nonionic surfactant has been replaced by an anionic one: sodium dodecyl sulfate (SDS). Changes induced by the addition of SDS have been determined by small-angle X-ray scattering (SAXS) and observations between crossed polarizers. Investigations have been carried out in both the water-rich and the oil-rich parts of the phase diagram, Changes in the Bragg distance, the Landau-Peierls order parameter eta and the elastic constants kappa and (B) over bar have been particularly examined. In the decane-swollen L-alpha phase, SAXS spectra did not present any changes. The Bragg distance d(B), as well as eta, remained constant as the nonionic surfactant was replaced by the anionic one. On the other hand, the water-swollen L-alpha phase was stiffened and,I was decreasing dramatically with charge addition. For the same membrane volume fraction, d(B) decreased slightly as SDS concentration increased. The dependence of the membrane rigidity kappa = KdB and the smectic compressibility modulus (B) over bar with the SDS concentration have been determined and compared with recent theoretical predictions. It was found that kappa increased with SDS concentration whereas (B) over bar presents a maximum when the Gouy-Chapman length is equal to the half of the water thickness.