Equilibrium molecular dynamics simulations are used to calculate the thermal conductivity of the one component plasma (OCP) via the Green-Kubo formalism over a broad range of Coulomb coupling strength, 0.1 ≤ Γ ≤ 180. These simulations address previous discrepancies between computations using equilibrium versus nonequilibrium methods. Analysis of heat flux autocorrelation functions show that very long (6 × 10 5 ω −1 p ) time series are needed to reduce the noise level to allow 2% accuracy. The new simulations provide the first accurate data for Γ 1. This enables a test of the traditional Landau-Spitzer theory, which is found to agree with the simulations for Γ 0.3. It also enables tests of theories to address moderate and strong Coulomb coupling. Two are found to provide accurate extensions to the moderate coupling regime of Γ 10, but none are accurate in the Γ 10 regime where potential energy transport and coupling between mass flow and stress dominate thermal conduction.