  We consider a branching-selection system of particles on the real line that
evolves according to the following rules: each particle moves according to a
Brownian motion during an exponential lifetime and then splits into two new
particles and, when a particle is at a distance $L$ of the highest particle, it
dies without splitting. This model has been introduced by Brunet, Derrida,
Mueller and Munier in the physics literature and is called the $L$-branching
Brownian motion. We show that the position of the system grows linearly at a
velocity $v_L$ almost surely and we compute the asymptotic behavior of $v_L$ as
$L$ tends to infinity: $v_L = \sqrt{2} - \pi^2 / 2 \sqrt{2} L^2 + o(1/L^2)$, as
conjectured by Brunet, Derrida, Mueller and Munier. The proof makes use of
results by Berestycki, Berestycki and Schweinsberg concerning branching
Brownian motion in a strip.
