  The scattering of charged solitons in the complex sine-Gordon field theory is
investigated. An exact factorizable S-matrix for the theory is proposed when
the renormalized coupling constant takes the values $\lambda^{2}_{R}=4\pi/k$
for any integer $k>1$: the minimal S-matrix associated with the Lie algebra
$a_{k-1}$. It is shown that the proposed S-matrix reproduces the leading
semiclassical behaviour of all amplitudes in the theory and is the minimal
S-matrix which is consistent with the semiclassical spectrum of the model. The
results are completely consistent with the description of the complex
sine-Gordon theory as the SU$(2)/{\rm U}(1)$ coset model at level $k$ perturbed
by its first thermal operator.
