  The fused six-vertex models with open boundary conditions are studied. The
Bethe ansatz solution given by Sklyanin has been generalized to the transfer
matrices of the fused models. We have shown that the eigenvalues of transfer
matrices satisfy a group of functional relations, which are the $su$(2) fusion
rule held by the transfer matrices of the fused models. The fused transfer
matrices form a commuting family and also commute with the quantum group
$U_q[sl(2)]$. In the case of the parameter $q^h=-1$ ($h=4,5,\cdots$) the
functional relations in the limit of spectral parameter $u\to \i\infty$ are
truncated. This shows that the $su$(2) fusion rule with finite level appears
for the six vertex model with the open boundary conditions. We have solved the
functional relations to obtain the finite-size corrections of the fused
transfer matrices for low-lying excitations. From the corrections the central
charges and conformal weights of underlying conformal field theory are
extracted. To see different boundary conditions we also have studied the
six-vertex model with a twisted boundary condition.
