  We study sequences of integral current spaces $(X_j,d_j,T_j)$ such that the
integral current structure $T_j$ has weight $1$ and no boundary and, all
$(X_j,d_j)$ are closed Alexandrov spaces with curvature uniformly bounded from
below and diameter uniformly bounded from above. We prove that for such
sequences either their limits collapse or the Gromov-Hausdorff and
Sormani-Wenger Intrinsic Flat limits agree. The latter is done showing that the
lower $n$ dimensional density of the mass measure at any regular point of the
Gromov-Hausdorff limit space is positive by passing to a filling volume
estimate.
  In an appendix we show that the filling volume of the standard $n$
dimensional integral current space coming from an $n$ dimensional sphere of
radius $r>0$ in Euclidean space equals $r^n$ times the filling volume of the
$n$ dimensional integral current space coming from the $n$ dimensional sphere
of radius $1$.
