A model for an ideally polarizable interface is proposed where the two sides of the interface are described by two interacting classical one component plasmas of different neutralizing background densities separated by an impermeable membrane. The statistical mechanics of this system is solved exactly in two dimensions at the reduced temperature 2. The one and two body distribution functions, the potential drop, the differential capacity, and the interfacial tension are computed. The main feature of this model is that the potential drop and not the surface charge appears as the natural external variable. Several sum rules are discussed, such as the screening theorems for multipoles, the contact theorem, and the Lippmann equation. A general thermodynamic argument shows that these sum rules remain valid in three dimensions and for any value of the coupling parameter.