  We consider the one-dimensional half-filled Hubbard model. We show that the
excitation spectrum is given by the scattering states of four elementary
excitations, which form the fundamental representation of $SU(2)\times SU(2)$.
We determine the exact two-particle Scattering matrix, which a solution of the
Yang-Baxter equation and reflects the $SO(4)$ symmetry of the model. The
results for repulsive and attractive Hubbard model are related by an
interchange of spin and charge degrees of freedom.
