We investigate the large N behavior of the smallest eigenvalue, lambda(N), of an (N + 1) x ( N + 1) Hankel (or moments) matrix H-N, generated by the weight w(x) = x(alpha)(1 - x)(beta), x is an element of [0, 1], alpha > 1, beta > 1. By applying the arguments of Szego, Widom and Wilf, we establish the asymptotic formula for the orthonormal polynomials P-n(z), z is an element of C \ [0, 1], associated with w(x), which are required in the determination of lambda(N). Based on this formula, we produce the expressions for lambda(N), for large N.