  This work concerns the boundary integrability of the
${\cal{U}}_{q}[osp(1|2)]$ Temperley-Lieb model. We constructed the solutions of
the graded reflection equations in order to determine the boundary terms of the
correspondig spin-1 Hamiltonian. We obtain the eigenvalue expressions as well
as its associated Bethe ansatz equations by means of the coordinate Bethe
ansatz. These equations provide the complete description of the spectrum of the
model with diagonal integrable boundaries.
