  The Nested Bethe Ansatz is generalized to open boundary conditions. This is
used to find the exact eigenvectors and eigenvalues of the $A_{n-1}$ vertex
model with fixed open boundary conditions and the corresponding $SU_{q}(n)$
invariant hamiltonian.
  The Bethe Ansatz equations obtained are solved in the thermodynamic limit
giving the vertex model free energy and the hamiltonian ground state energy
including the corresponding boundary contributions.
