  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\"{\i}d\'ekon (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.
