  We derive the exact general form of the equation of state, in the fugacity
format, for the two-dimensional Coulomb gas. Our results are valid in the
conducting phase of the Coulomb gas, for temperatures above the
Kosterlitz-Thouless transition. The derivation of the equation of state is
based on the knowledge of the general form of the short-distance expansion of
the correlation functions of the Coulomb gas. We explicitly compute the
expansion up to order $O(\zeta^6)$ in the activity $\zeta$. Our results are in
very good agreement with Monte Carlo simulations at very low density.
