We revisit the equilibrium classical statistical mechanics of the two-dimensional one-component plasma, for the special value Г = 2 of the coupling constant. Using a new method, we find that the model is solvable (the n-body densities can be explicitly computed) for a larger class of inhomogeneous backgrounds. In particular, we can deal with a doubly periodic background; this is a classical model for a crystal made of fixed ions and mobile electrons. At Г = 2, this system is conducting: the correlations have a fast decay, and the Stillinger-Lovett screening sum rule is obeyed.