In this work, we report results for the phase diagram of the two-dimensional Coulomb system. We use a recently proposed simulation method, thermodynamic scaling Monte Carlo, to determine the Kosterlitz-Thouless insulator/conductor transition line and the first-order vapor-liquid coexistence curve. The Kosterlitz-Thouless line terminates at the first-order coexistence curve close to the gas-liquid critical point. Our estimate of the vapor-liquid critical point is, T*approximate to 0.056, rho*approximate to 0.21, in only modest agreement with previous theoretical estimates and semiquantitative numerical results.