  The complete spectrum of states in the supersymmetric principal chiral model
based on SU(n) is conjectured, and an exact factorizable S-matrix is proposed
to describe scattering amongst these states. The SU(n)_L*SU(n)_R symmetry of
the lagrangian is manifest in the S-matrix construction. The supersymmetries,
on the other hand, are incorporated in the guise of spin-1/2 charges acting on
a set of RSOS kinks associated with su(n) at level n. To test the proposed
S-matrix, calculations of the change in the ground-state energy in the presence
of a coupling to a background charge are carried out. The results derived from
the lagrangian using perturbation theory and from the S-matrix using the TBA
are found to be in complete agreement for a variety of background charges which
pick out, in turn, the highest weight states in each of the fundamental
representations of SU(n). In particular, these methods rule out the possibility
of additional CDD factors in the S-matrix. Comparison of the expressions found
for the free-energy also yields an exact result for the mass-gap in these
models: m/Lambda_{MS-bar}=(n/pi)sin(pi/n).
