We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift $c$. At the point $x > 0$, we add an absorbing barrier, i.e.\ individuals touching the barrier are instantly killed without producing offspring. It is known that there is a critical drift $c_0$, such that this process becomes extinct almost surely if and only if $c \ge c_0$. In this case, if $Z_x$ denotes the number of individuals absorbed at the barrier, we give an asymptotic for $P(Z_x=n)$ as $n$ goes to infinity. If $c=c_0$ and the reproduction is deterministic, this improves upon results of L. Addario-Berry and N. Broutin (2011) and E. A\"��d��kon (2010) on a conjecture by David Aldous about the total progeny of a branching random walk. The main technique used in the proofs is analysis of the generating function of $Z_x$ near its singular point 1, based on classical results on some complex differential equations.

31 pages, final version, to appear in Annales de l'Institut Henri Poincar\'e, Section B. Corrects an error in proof of Theorem 1.1 and adds reference to Yang and Ren(2011)