General formulas for the ordinary and Soret diffusion coefficients D-12 and D-T, and the thermal-diffusion ratio kr (or the thermal-diffusion factor alpha(12)) of a binary and dilute fluid mixture consisting of nonspherical molecules have been obtained utilizing the classical transport theory recently developed by us [G. S. Singh and B. Kumar, J. Chem. Phys. 104, 5604 (1996)]. The general results involve up to the infinite order of approximations but in the present work we have restricted ourselves up to the second order only. We thus obtain general expressions for the second approximation to both D-12 and D-T, and the first approximation to k(T) (or alpha(12)) in terms of different square bracket integrals. The velocity as well as angular velocity parts of these integrals for the fluid mixtures of hard biaxial ellipsoids have been analytically performed with the help of the modified Hoffman procedure. The expressions for [D-12]2, [D-T](2), and [k(T)](1) have thus ultimately been obtained in terms of five four-dimensional quadratures over the orientational coordinates of like or unlike pairs of colliding ellipsoidal molecules.