  The interaction potential between supported floating bilayers has been
determined by grazing incidence specular and off-specular scattering using
synchrotron radiation. Our measurements demonstrate that floating bilayers are
significantly more hydrated than the usually studied multilayers, therefore
giving access to more intrinsic properties of the membranes. At equilibrium,
the interaction potential can be two orders of magnitude softer than previously
reported, which can be explained by the weak electrostatic interaction due to
the small fraction (sigma ~ 0.001 e$^-$/nm$^2$) of ionized lipids in the
bilayers. Our data are sensitive enough to discriminate between different
functional forms of the entropic repulsion proposed in the literature, and the
"soft" potential of R. Podgornik and V. Parsegian, Langmuir 8, 557 (1992) is
shown to most appropriately describe our data.
