The scattering of low-energy electrons by polar molecules is considered. The excitation to the rotational states in the sudden approximation and the effect of the Coulomb dipole force to all orders are taken into account. Analytic expressions for the scattering amplitudes are obtained. These are valid for arbitrarily large dipole moments of the molecule. Differential and momentum-transfer cross sections have been calculated for several molecules and compared with experimental data, with the Born approximation and with coupled-channels calculations. © 1998 The American Physical Society.