The thermodynamic limit of the excess internal energy per particle u of the fluid phase of the three-dimensional one-component plasma is investigated by means of Monte Carlo simulations. The simulations are performed in the canonical ensemble within hyperspherical boundary conditions. u is computed for 31 values of the coupling parameter Gamma in the range 1 less than or equal to Gamma less than or equal to 190. For each value of Gamma the thermodynamic limit of u is obtained by studying the scaling law which governs the behavior of u with the number N of particles.