Abstract. Galton–Watson branching processes with immigration are considered, in which the mathematical expectation A of the number of offspring of a single individual tends towards unity. For the process denoting the number of individuals that existed before generation n (total progeny), normalized by mathematical expectation, an asymptotic distribution has been found that depends on the rate at which A tends towards unity.

Keywords: branching processes with immigration, total progeny, transient phenomena.words: branching processes with immigration, total progeny, transient phenomena.