  The recently introduced two-parameter eight-state $U_q[gl(3|1)]$
supersymmetric fermion model is extended to include boundary terms. Nine
classes of boundary conditions are constructed, all of which are shown to be
integrable via the graded boundary quantum inverse scattering method. The
boundary systems are solved by using the coordinate Bethe ansatz and the Bethe
ansatz equations are given for all nine cases.
