We study the large time behaviour of the reaction-diffsuion equation ∂ t u = ∆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 ∞ of the unit sphere, such that u(t, x) converges uniformly in R N , as t goes to infinity, to U c * |x| − c * t + N − 1 c * lnt + s ∞ x |x| , 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→+∞ (t) precision, or identified precisely the level sets locations for almost radial initial data.