In this paper, we study the large N behavior of the smallest eigenvalue λ of the (N+1)×(N+1) Hankel matrix, H=(μ), generated by the γ dependent Jacobi weight w(z,γ)=ez(1−z),z∈[0,1],γ∈R,α>−1,β>−1. Applying the arguments of Szegö, Widom and Wilf, we obtain the asymptotic representation of the orthonormal polynomials P(z), z∈C﹨[0,1], with the weight w(z,γ)=ez(1−z). Using the polynomials P(z), we obtain the theoretical expression of λ, for large N. We also display the smallest eigenvalue λ for sufficiently large N, computed numerically.