A numerical study of heat and mass transfer in two-dimensional stagnation-point flow of an incompressible viscous fluid over a stretching vertical sheet in the present of thermal-diffusion (Soret) and diffusion-thermo (Dufour) numbers is investigated. The analysis accounts for both assisting and opposing flows and temperature dependent thermal conductivity. The set of governing equations and the boundary condition are reduced to ordinary differential equations with appropriate boundary conditions. Furthermore, the similarity equations are solved numerically by using fourth order Runge-Kutta integration scheme with Newton Raphson shooting method. The accuracy of the numerical method is tested by performing various comparisons with previously published work and the results are found to be in excellent agreement. Numerical results for velocity, temperature and concentration distributions as well as skin friction coefficient, local Nusselt number and local Sherwood numbers are discussed for various values of physical parameters.