It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn } exists such that {Xn/cn } converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 < m < ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn } such that {Xn /c n} converges in law to a proper limit distribution function F, with F(0 +) < 1.