  The Aharonov-Bohm-Coulomb potentials in two dimensions may describe the
interaction between two particles carrying electric charge and magnetic flux,
say, Chern--Simons solitons, or so called anyons. The scattering problem for
such two-body systems is extended to the relativistic case, and the scattering
amplitude is obtained as a partial wave series. The electric charge and
magnetic flux is ($-q$, $-\phi/Z$) for one particle and ($Zq$, $\phi$) for the
other. When $(Zq^2/\hbar c)^2\ll 1$, and $q\phi/2\pi\hbar c$ takes on integer
or half integer values, the partial wave series is summed up approximately to
give a closed form. The results exhibit some nonperturbative features and
cannot be obtained from perturbative quantum electrodynamics at the tree level.
