  We present an approximation to the Brunet--Derrida model of supercritical
branching Brownian motion on the real line with selection of the $N$ right-most
particles, valid when the population size $N$ is large. It consists of
introducing a random space-time barrier at which particles are instantaneously
killed in such a way that the population size stays almost constant over time.
We prove that the suitably recentered position of this barrier converges at the
$\log^3 N$ timescale to a L\'evy process, which we identify. This validates the
physicists' predictions about the fluctuations in the Brunet--Derrida model.
