It is shown that when the Born approximation is applied to rearrangement collisions in the customary way, terms of the first order in the interaction energy between the colliding particles are omitted from the exchange scattering amplitude. If these terms are retained the arbitrariness which arises from the lack of orthogonality between the initial and final state wave functions is removed. The first-order exchange approximation derived in the present paper is employed to calculate the cross-sections for the 1 s -2 s and 1 s -2 p excitations of hydrogen atoms by electron impact and the elastic scattering of electrons by hydrogen atoms.