Let a and b be nonnegative integers with 2 $leq$ a < b, and let G be a Hamiltonian graph of order n with n > $frac{(a+b-5)(a+b-3)}{b-2}$. An [a, b]-factor F of G is called a Hamiltonian [a, b]-factor if F contains a Hamiltonian cycle. In this paper, it is proved that G has a Hamiltonian [a, b]-factor if $delta(G);geq;frac{(a-1)n+a+b-3)}{a+b-3}$ and $delta(G)$ > $frac{(a-2)n+2{alpha}(G)-1)}{a+b-4}$.