We study a Coulomb gas on a sufficiently smooth Jordan arc in the complex plane, at arbitrary positive temperature. We show that, as the number of particles tends to infinity, the partition function converges to an expression involving the partition function of the gas on [−1,1], a power of the capacity of the curve, and the Fredholm determinant of the arc-Grunsky operator. We also obtain an asymptotic formula for the Laplace transform of linear statistics for sufficiently regular test functions. This shows that the centered empirical measure converges to a Gaussian field with explicit asymptotic mean and variance given by the Dirichlet energy of the test function. Based on joint work with Kurt Johansson and Fredrik Viklund.