  An accurate equation of state of the one component plasma is obtained in the
low coupling regime $0 \leq \Gamma \leq 1$. The accuracy results from a smooth
combination of the well-known hypernetted chain integral equation, Monte Carlo
simulations and asymptotic analytical expressions of the excess internal energy
$u$. In particular, special attention has been brought to describe and take
advantage of finite size effects on Monte Carlo results to get the
thermodynamic limit of $u$. This combined approach reproduces very accurately
the different plasma correlation regimes encountered in this range of values of
$\Gamma$. This paper extends to low $\Gamma$'s an earlier Monte Carlo
simulation study devoted to strongly coupled systems for $1 \leq \Gamma \leq
190$ ({J.-M. Caillol}, {J. Chem. Phys.} \textbf{111}, 6538 (1999)). Analytical
fits of $u(\Gamma)$ in the range $0 \leq \Gamma \leq 1$ are provided with a
precision that we claim to be not smaller than $p= 10^{-5}$. HNC equation and
exact asymptotic expressions are shown to give reliable results for $u(\Gamma)$
only in narrow $\Gamma$ intervals, i.e. $0 \leq \Gamma \lesssim 0.5$ and $0
\leq \Gamma \lesssim 0.3$ respectively.
