The authors prove the following theorem. Let \(M^ n\) be a hypersurface of an \((n+1)\)-dimensional Riemannian manifold \(S^{n+1}(c)\) with constant curvature. Then \(M^ n\) has constant scalar curvature R and is minimal if and only if (1) when \(R=n(n-1)c\), \(M^ n\) is totally geodesic, (2) when \(R\neq n(n-1)c\), \(M^ n\) is locally decomposable and \(M^ n=R^ 1\times S^{n-1}(k)\), where \(k=nc/(n-1)\) and \(c>0\). From this theorem, they obtain the following corollary: If \(M^ n\) (n\(\geq 4)\) is conformally flat and minimal with constant scalar curvature, then it is totally geodesic.