  We study stability of conservative solutions of the Cauchy problem for the
periodic Camassa-Holm equation $u_t-u_{xxt}+\kappa
u_x+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial data $u_0$. In particular, we
derive a new Lipschitz metric $d_\D$ with the property that for two solutions
$u$ and $v$ of the equation we have $d_\D(u(t),v(t))\le e^{Ct} d_\D(u_0,v_0)$.
The relationship between this metric and usual norms in $H^1_{\rm per}$ and
$L^\infty_{\rm per}$ is clarified.
