  We propose that the Baxter's $Q$-operator for the XYZ quantum spin chain with
open boundary conditions is given by the $j\to \infty$ limit of the
corresponding transfer matrix with spin-$j$ (i.e., $(2j+1)$-dimensional)
auxiliary space. The associated $T$-$Q$ relation is derived from the fusion
hierarchy of the model. We use this relation to determine the Bethe Ansatz
solution of the eigenvalues of the fundamental transfer matrix. This solution
yields the complete spectrum of the Hamiltonian.
