A blow-up result for the Cauchy problem for the periodic Camassa-Holm equation is given. Namely, it is proved that if the initial value \(u_0\in H^4(S)\), \(S=\mathbb{R}/\mathbb{Z}\), has at some point the slope less than \(-\sqrt{13/12}|u_0|_{H^1(S)}\), then the solution blows-up in finite time \(T\). The solution remains bounded in \([0;T)\), but its slope is unbounded in the neighbourhood of \(T\).