Using the method of compensated compactness, the authors prove the existence of a global weak solution to the initial value problem for the Camassa-Holm equation \(u_t- u_{xxt}+ 3uu_x= 2u_x u_{xx}+ uu_{xxx}\), \(t>0\), \(x\in\mathbb{R}\), \(u(0, x)= u_0(x)\in H^1(\mathbb{R})\). This problem describes a unidirectional propagation of water waves on a free surface, and is capable of treating the interaction of peaked solutions (solitons with cusp singularities).