  We diagonalise the Hamiltonian of the Temperley-Lieb loop model with open
boundaries using a coordinate Bethe Ansatz calculation. We find that in the
groundstate sector of the loop Hamiltonian, but not in other sectors, a certain
constraint on the parameters has to be satisfied. This constraint has a natural
interpretation in the Temperley-Lieb algebra with boundary generators. The
spectrum of the loop model contains that of the quantum spin-1/2 XXZ chain with
non-diagonal boundary conditions. We hence derive a recently conjectured
solution of the complete spectrum of this XXZ chain. We furthermore point out a
connection with recent results for the two-boundary sine-Gordon model.
