We study stability of solutions of the Cauchy problem for the Hunter–Saxton equation ut + uux = 14 ( R x −∞ u 2 x dx− R∞ x ux dx) with initial data u0. In particular, we derive a new Lipschitz metric dD with the property that for two solutions u and v of the equation we have dD(u(t), v(t)) ≤ edD(u0, v0).