  We have recently constructed a large class of open quantum spin chains which
have quantum-algebra symmetry and which are integrable. We show here that these
models can be exactly solved using a generalization of the analytical Bethe
Ansatz (BA) method. In particular, we determine in this way the spectrum of the
transfer matrices of the $U_q [(su(2)]$-invariant spin chains associated with
$A^{(1)}_1$ and $A^{(2)}_2$ in the fundamental representation. The
quantum-algebra invariance of these models plays an essential role in obtaining
these results. The BA equations for these open chains are ``doubled'' with
respect to the BA equations for the corresponding closed chains.
