It is shown that the limiting random variable W(si ) of an irregular branching process with infinite mean, defined in [5], has a continuous and positive distribution on {0 < W(si ) < ∞}. This implies that for all branching processes (Zn ) with infinite mean there exists a function U such that the distribution of V = limn U(Zn )e–n a.s. is continuous, positive and finite on the set of non-extinction. A kind of law of large numbers for sequences of independent copies of W(si ) is derived.