We investigate the large N behavior of the smallest eigenvalue, λ N , of an (N + 1) × (N + 1) Hankel (or moments) matrix H N , generated by the weight w(x) = x α (1 − x) β , x ∈ [0, 1], α > −1, β > −1. By applying the arguments of Szegö, Widom and Wilf, we establish the asymptotic formula for the orthonormal polynomials P n (z), z ∈ C \ [0, 1], associated with w(x), which are required in the determination of λ N . Based on this formula, we produce the expressions for λ N , for large N .Using the parallel algorithm presented by Emmart, Chen and Weems, we show that the theoretical results are in close proximity to the numerical results for sufficiently large N .