Let V be a codimension one integral varifold with bounded anisotropic mean curvature such that ℋⁿ(spt‖V‖ ∩ K) < ∞ for K compact. Some days ago we (me and Mario) realised that our paper actually proves almost everywhere regularirty for V. The estimates for the Minkowski content of a compact subset K of an (n,h)-set (see 3.4 in paper) may be upgraded to a precise formula that expresses the Minkowski content of K in terms of the Hausdorff measure of the part of K that can be touched by exactly one or two balls. Moreover, if K is countably (ℋⁿ,n) rectifiable, then the measure of the part of K that can be touched by exactly one ball is zero. Hence, we get quadratic hight decay at almost all points of K. Since spt‖V‖ is an (n,h)-set this allows to apply Allard’s regularity theorem ‖V‖ almost everywhere.