The diffusion-slip problem for a binary mixture of gases is investigated on the basis of the linearized Boltzmann equation for hard-sphere molecules with the diffuse reflection boundary condition. The problem is analyzed numerically by the finite-difference method, where the collision integrals are computed by the numerical kernel method first introduced by Sone, Ohwada and Aoki for one-component gases [Sone et. al., Phys. Fluids A, Vol. 1,363 (1989)]. This is the first report in which the method is extended and applied to the case of mixtures. The analysis is carried out for several combinations of the component gases and the behavior of the mixture is clarified at the level of the velocity distribution functions. As a result, the coefficient of the diffusion-slip and the associated Knudsen-layer functions are obtained.