The purpose of this paper is to give an estimate of the error in approximating the integral $\int\limits_{-\infty}^\infty f(x)\exp(-x^2)dx$ by the Gauss-Jacobi quadrature formula $Q_n(w;f)$, assuming that $f$ is an entire function satisfying a certain growth condition which depends on the Hermite weight function $w(x)= \exp(-x^2)$.