  We study the conformal field theory of a free massless scalar field living on
the half line with interactions introduced via a periodic potential at the
boundary. An SU(2) current algebra underlies this system and the interacting
boundary state is given by a global SU(2) rotation of the left-moving fields in
the zero-potential (Neumann) boundary state. As the potential strength varies
from zero to infinity, the boundary state interpolates between the Neumann and
the Dirichlet values. The full S-matrix for scattering from the boundary, with
arbitrary particle production, is explicitly computed. To maintain unitarity,
it is necessary to attribute a hidden discrete ``soliton'' degree of freedom to
the boundary. The same unitarity puzzle occurs in the Kondo problem, and we
anticipate a similar solution.
