We consider branching Brownian motion on the real line with the following selection mechanism: Every time the number of particles exceeds a (large) given number N , only the N right-most particles are kept and the others killed. After rescaling time by log 3 N , we show that the properly recentred position of the rαN s-th particle from the right, α P p0, 1q, converges in law to an explicitly given spectrally positive Lévy process. This behaviour has been predicted to hold for a large class of models falling into the universality class of the FKPP equation with weak multiplicative noise [Brunet et al., Phys. Rev. E 73(5), 056126 (2006)] and is proven here for the first time for such a model.