We demonstrate a quasipolynomial-time deterministic approximation algorithm for the partition function of a Gibbs point process interacting via a finite-range stable potential. This result holds for all activities $λ$ for which the partition function satisfies a zero-free assumption in a neighborhood of the interval $[0,λ]$. As a corollary, for all finite-range stable potentials we obtain a quasipolynomial-time determinsitic algorithm for all $λ< /(e^{B + 1} \hat C_φ)$ where $\hat C_φ$ is a temperedness parameter and $B$ is the stability constant of $φ$. In the special case of a repulsive potential such as the hard-sphere gas we improve the range of activity by a factor of at least $e^2$ and obtain a quasipolynomial-time deterministic approximation algorithm for all $λ< e/Δ_φ$, where $Δ_φ$ is the potential-weighted connective constant of the potential $φ$. Our algorithm approximates coefficients of the cluster expansion of the partition function and uses the interpolation method of Barvinok to extend this approximation throughout the zero-free region.