  The open XXZ spin chain with the anisotropy $\Delta=-\frac12$ and a
one-parameter family of diagonal boundary fields is studied at finite length. A
determinant formula for an overlap involving the spin chain's ground-state
vectors for different lengths is found. The overlap allows one to obtain an
exact finite-size formula for the ground state's logarithmic bipartite
fidelity. The leading terms of its asymptotic series for large chain lengths
are evaluated. Their expressions confirm the predictions of conformal field
theory for the fidelity.
