  We study convergence of the spectral shift function for the finite interval
restrictions of a pair of full-line Schr\"odinger operators to an interval of
the form $(-\ell,\ell)$ with coupled boundary conditions at the endpoints as
$\ell\to \infty$ in the case when the finite interval restrictions are
relatively prime to those with Dirichlet boundary conditions. Using a
Krein-type resolvent identity we show that the spectral shift function for the
finite interval restrictions converges weakly to that for the pair of full-line
Schr\"odinger operators as the length of the interval tends to infinity.
