  We study stability of solutions of the Cauchy problem for the Hunter-Saxton
equation $u_t+uu_x=\frac14(\int_{-\infty}^xu_x^2 dx-\int_{x}^\infty u_x^2 dx)$
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
$d_\D(u(t),v(t))\le e^{Ct} d_\D(u_0,v_0)$.
