  We study the manner in which spectral shift functions associated with
self-adjoint one-dimensional Schr\"odinger operators on the finite interval
$(0,R)$ converge in the infinite volume limit $R\to\infty$ to the half-line
spectral shift function.
  Relying on a Fredholm determinant approach combined with certain measure
theoretic facts, we show that prior vague convergence results in the literature
in the special case of Dirichlet boundary conditions extend to the notion of
weak convergence and arbitrary separated self-adjoint boundary conditions at
$x=0$ and $x=R$.
