||We consider one-dimensional
branching Brownian motion in which particles are absorbed at the origin. We assume that when a particle
branches, the offspring distribution is supercritical, but the particles are given a
critical drift towards the origin so that the process eventually goes extinct with probability one. We establish precise asymptotics for the probability that
the process survives for a large time t, improving upon a result of Kesten
(1978) and Berestycki, Berestycki, and Schweinsberg (2014). We also prove a Yaglom-type limit theorem for the behavior of the process conditioned to survive for an unusually long time, which also improves upon results of
Kesten (1978). An important tool in the proofs of these results is the convergence of branching Brownian motion with absorption to a continuous state branching process.