  We find the exact quasiparticle spectrum for the continuum Kondo problem of
$k$ species of electrons coupled to an impurity of spin $S$. In this
description, the impurity becomes an immobile quasiparticle sitting on the
boundary. The particles are ``kinks'', which can be thought of as field
configurations interpolating between adjacent wells of a potential with $k+1$
degenerate minima. For the overscreened case $k>2S$, the boundary has this kink
structure as well, which explains the non-integer number of boundary states
previously observed. Using simple arguments along with the consistency
requirements of an integrable theory, we find the exact elastic $S$-matrix for
the quasiparticles scattering among themselves and off of the boundary. This
allows the calculation of the exact free energy, which agrees with the known
Bethe ansatz solution.
