  We study the large time behaviour of the reaction-diffsuion equation
$\partial_t u=\Delta u +f(u)$ in spatial dimension $N$, when the nonlinear term
is bistable and the initial datum is compactly supported. We prove the
existence of a Lipschitz function $s^\infty$ of the unit sphere, such that
$u(t,x)$ converges uniformly in $\mathbb{R}^N$, as $t$ goes to infinity, to
  $U_{c_*}\bigg(|x|-c_*t + \frac{N-1}{c_*} \mathrm{ln}t +
s^\infty\Big(\frac{x}{|x|}\Big)\bigg)$, where $U_{c*}$ is the unique 1D
travelling profile. This extends earlier results that identified the locations
of the level sets of the solutions with $o_{t\to+\infty}(t)$ precision, or
identified precisely the level sets locations for almost radial initial data.
