An analytical solution of the Ornstein-Zernike equation for a multicomponent hard-sphere fluid is studied with the closure of the usual hard-sphere condition and the following:   where c_ij(r) is the direct correlation function, σ_ij is the closest distance between the i and j species of spheres and σ*_ij = σ_ij + 0⁺ with a positive infinitesimal 0⁺. In the case when the coefficients Λ_ij σ²_ij and K_ij are factorizable, the full analytical solution in particular is obtained explicitly in terms of rational functions of the parameter which is the physical solution of the nonlinear equation.