Italians and Irish have been emigrating for centuries which is part of the reason why the birthrate in Italy is less than the replacementrate because who wants to stay in a place where there is little opportunity and there are no jobs?
The natural birthrate in the USA has been below natural replacementrate for nearly a generation, and the USA has relied on massive immigration to continue population growth.
This could be achieved through a cap-and-trade system by issuing each adult with 1.05 of a birth permit (ie, 2.1 permits per couple to achieve the replacement fertility rate) and allowing such permits to be tradable.