  The model under consideration is the two-dimensional Coulomb gas of $\pm$
charged hard disks with diameter $\sigma$. For the case of pointlike charges
$(\sigma=0)$, the system is stable against collapse of positive-negative pairs
of charges in the range of inverse temperatures $0 \le \beta < 2$, where its
full thermodynamics was obtained exactly [L. {\v{S}}amaj and I. Trav{\v{e}}nec,
{\it J. Stat. Phys.} {\bf 101}:713 (2000)]. In the present work, we derive the
leading correction to the exact thermodynamics of pointlike charges due to the
presence of the hard core $\sigma$ (appearing in the dimensionless combination
$n\sigma^2$, $n$ is the particle density). This permits us to extend the
treatment to the interval $2\le \beta <3$ (the Kosterlitz-Thouless phase
transition takes place at $\beta=4$). The results, which are exact in the
low-density limit $n\sigma^2 \to 0$, reproduce correctly the singularities of
thermodynamic quantities at the collapse point $\beta=2$ and agree very well
with Monte-Carlo simulations.
