The distorted wave Born series is applied to the problem of rotational inelastic scattering of an atom by a diatomic molecule. The formulation is made in the total angular momentum representation, and for the special but important case of an exponential interaction a simple second-order expression for the degeneracy averaged cross section is derived. Numerical results for two intermolecular potentials compare favorably with exact and DW results do not vanish near threshold, (2) show the applicability of the method to molecules heavier than H2, (3) provide a method of assessing the relative order of contributions from various internal states, (4) provide a method of estimating higher order terms in the DWB series. The application of the second-order DWB series to problems with different forms of potentials of interaction requires only a single numerical integration. (Author).