  The paper provides a direct proof the uniqueness of solutions to the
Camassa-Holm equation, based on characteristics. Given a conservative solution
$u=u(t,x)$, an equation is introduced which singles out a unique characteristic
curve through each initial point. By studying the evolution of the quantities
$u$ and $v= 2\arctan u_x$ along each characteristic, it is proved that the
Cauchy problem with general initial data $u_0\in H^1(\mathbb{R})$ has a unique
solution, globally in time.
