Let \(M_{1}^{n+1}(c)\), \(n\geq 3\), be an \((n+1)\)-dimensional connected indefinite Riemannian manifold of index 1 and of constant curvature \(c\). According to \(c > 0\), \(c=0\) and \(c<0\), it is called de Sitter space, Minkowski space or anti-de Sitter space, respectively, and it is denoted by \(S_{1}^{n+1}(c)\), \({\mathbb R}_{1}^{n+1}(c)\) or \(H_{1}^{n+1}(c)\). A hypersurface \(M\) of \(M_{1}^{n+1}(c)\) is said to be spacelike if the metric induced on \(M\) from the ambient space is positive definite. In the paper under review, it is shown that on any compact spacelike hypersurface \(M\) of de Sitter space \(S_{1}^{n+1}(c)\), a certain integral equality is fulfilled. Using this fact, the authors prove that if the hypersurface \(M\) has constant scalar curvature \(\kappa \), \(\kappa = n(n-1)r\) and \(r < c\), then \(M\) is isometric to a sphere.