General expressions in terms of the square bracket integrals for the shear and volume viscosities of a classical dilute binary gas mixture of nonspherical molecules have been derived by extending the Taxman classical kinetic theory of transport phenomena within the framework of the Chapman-Enskog second approximation method for a mixture of spherical molecules. The dimensionality of integrals for the binary mixture of rigid spheroidal molecules having C-infinity upsilon symmetry has been reduced analytically to get ultimately the forms wherein the integrals are to be performed only over the orientational coordinates of two colliding like or unlike spheroids. The general expressions for the nonspherical molecules as well as the final integrals for the spheroidal molecules are shown to be correct. The shear- and volume-viscosity formulas for a classical dilute binary gas mixture of hard sphere molecules have been retrieved as the limiting case of our corresponding expressions for the spheroidal systems.