  In a previous paper, the authors proved a conjecture of Lalley and Sellke
that the empirical (time-averaged) distribution function of the maximum of
branching Brownian motion converges almost surely to a Gumbel distribution. The
result is extended here to the entire system of particles that are extremal,
i.e. close to the maximum. Namely, it is proved that the distribution of
extremal particles under time-average converges to a Poisson cluster process.
