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We study some aspects of the equilibrium statistical mechanics of a classical two-dimensional one-component Coulomb plasma (the interaction is logarithmic). The free energy is calculated exactly, for one special value of the temperature. At low temperature, the particles form a triangular lattice, which is stable ; its vibration modes are well-behaved longitudinal plasmons and transverse phonons. The self-consistent harmonic approximation is studied ; it has a solution for the transverse sound velocity at low temperature only, a strong indication that the solid phase is unstable at high temperature. The free énergies obtained in different ways are compared. It is very likely that the model has a solid-fluid phase transition