  We examine an electric double layer containing an antagonistic salt in an
aqueous mixture, where the cations are small and hydrophilic but the anions are
large and hydrophobic. In this situation, a strong coupling arises between the
charge density and the solvent composition. As a result, the anions are trapped
in an oil-rich adsorption layer on a hydrophobic wall. % while the cations are
expelled from it. We then vary the surface charge density $\sigma$ on the wall.
For $\sigma>0$ the anions remain accumulated, but for $\sigma<0$ the cations
are attracted to the wall with increasing $|\sigma|$. Furthermore, the electric
potential drop $\Psi(\sigma)$ is nonmonotonic when the solvent interaction
parameter $\chi(T)$ exceeds a critical value $\chi_c$ determined by the
composition and the ion density in the bulk. This leads to a first order phase
transition between two kinds of electric double layers with different $\sigma$
and common $\Psi$. In equilibrium such two layer regions can coexist. The
steric effect due to finite ion sizes is crucial in these phenomena.
