  We show that the Cauchy problem for the Camassa-Holm equation has a unique,
global, weak, and dissipative solution for any initial data $u_0\in
H^1(\mathbb{R})$, such that $u_{0,x}$ is bounded from above almost everywhere.
In particular, we establish a one-to-one correspondence between the properties
specific to the dissipative solutions and a solution operator associating to
each initial data exactly one solution.
