  In this paper, we study locally strongly convex affine hypersurfaces with
vanishing Weyl curvature tensor and semi-parallel cubic form relative to the
Levi-Civita connection of affine metric. As the main result, we classify such
hypersurfaces being not of flat affine metric. In particular, $2,
3$-dimensional locally strongly convex affine hypersurfaces with semi-parallel
cubic form are completely determined.
