The equilibrium distance between two lipid bilayers stable in bulk water and in proximity of a substrate was investigated. Samples consisted of a homogeneous lipid bilayer, floating near an identical bilayer deposited on the hydrophilic surface of a silicon single crystal. Lipids were saturated diacyl phosphocholines, with the number of carbon atoms per chain, n, varying from 16 to 20. The average and root-mean-square positions of the floating bilayer were determined by means of neutron specular reflectivity. Samples were prepared at room temperature (i.e., with the lipids in the gel phase), and measurements were performed at various temperatures so that the whole region of transition from gel to fluid phase was explored. Data have been interpreted in terms of competition between the interbilayer potential and membrane fluctuations and used to estimate the bending rigidity of the bilayer.