  We consider a (one-dimensional) branching Brownian motion process with a
general offspring distribution having at least two moments, and in which all
particles have a drift towards the origin where they are immediately absorbed.
It is well-known that if and only if the branching rate is sufficiently large,
then the population survives forever with positive probability. We show that
throughout this super-critical regime, the number of particles inside any fixed
set normalized by the mean population size converges to an explicit limit,
almost surely and in $L^1$. As a consequence, we get that almost surely on the
event of eternal survival, the empirical distribution of particles converges
weakly to the (minimal) quasi-stationary distribution associated with the
Markovian motion driving the particles. This proves a result of Kesten from
1978, for which no proof was available until now.
