Barbosa, do Carmo and Eschenburg characterized the totally umbilical spheres as the only weakly stable compact constant mean curvature hypersurfaces in the Euclidean sphere $\mathbb {S}^{n+1}$. In this paper we prove that the weak index of any other compact constant mean curvature hypersurface $M^n$ in $\mathbb {S}{n+1}$ which is not totally umbilical and has constant scalar curvature is greater than or equal to $n+2$, with equality if and only if $M$ is a constant mean curvature Clifford torus $\mathbb {S}^{k}(r)\times \mathbb {S}^{n-k}(\sqrt {1-r^2})$ with radius $\sqrt {k/(n+2)}\leqslant r\leqslant \sqrt {(k+2)/(n+2)}$.