For an immersed hypersurface \documentclass[12pt]{minimal}
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\begin{document}$${f : M^n \rightarrow R^{n+1}}$$\end{document} without umbilical points, one can define the Möbius metric g on f which is invariant under the Möbius transformation group. The volume functional of g is a generalization of the well-known Willmore functional, whose critical points are called Willmore hypersurfaces. In this paper, we prove that if a n-dimensional Willmore hypersurfaces \documentclass[12pt]{minimal}
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\begin{document}$${(n \geq 3)}$$\end{document} has constant sectional curvature c with respect to g, then c = 0, n = 3, and this Willmore hypersurface is Möbius equivalent to the cone over the Clifford torus in \documentclass[12pt]{minimal}
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\begin{document}$${S^{3} \subset R^{4}}$$\end{document} . Moreover, we extend our previous classification of hypersurfaces with constant Möbius curvature of dimension \documentclass[12pt]{minimal}
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\begin{document}$${n \ge 4}$$\end{document} to n = 3, showing that they are cones over the homogeneous torus \documentclass[12pt]{minimal}
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\begin{document}$${S^1(r) \times S^1(\sqrt{1 - r^2}) \subset S^3}$$\end{document} , or cylinders, cones, rotational hypersurfaces over certain spirals in the space form R2, S2, H2, respectively.
