In this paper, we derive a precise estimate for the mean extinction time of the contact process with a fixed infection rate on a star graph with $N$ leaves. Specifically, we determine not only the exponential main factor but also the exact sub-exponential prefactor in the asymptotic expression for the mean extinction time as $N\to\infty$. Previously, such detailed asymptotic information on the mean extinction time of the contact process was available exclusively for complete graphs. To obtain our results, we first establish an accurate estimate for the stationary distribution of a modified contact process, employing special function theory and refined Laplace's method. Subsequently, we apply a recently developed potential theoretic approach for analyzing metastability in non-reversible Markov processes, enabling us to deduce the asymptotic expression. The integration of these methodologies constitutes a novel approach developed in this paper, one which has not been utilized previously in the study of the contact process.