  We prove that a stable minimal hypersurface of an open ball having a singular
set of locally finite codimension 2 Hausdorff measure which is weakly close to
a multiplicity 2 hyperplane is a 2-valued C^{1, alpha} graph in the interior.
Applications including a compactness theorem for a class of immersed stable
minimal hypersurfaces and a pointwise curvature estimate for the hypersurfaces
in this class in low dimensions are also discussed.
