In the present paper we continue the study started in [6] of semigroups whose power semigroups are Archimedean. We prove that these semigroups are exactly the nilpotent extensions of rectangular bands, and that the class of all such semigroups is power closed. Some special kinds of Archimedean semigroups will be described in terms of their forbidden divisors. We also outline some consequences of the obtained results which concern the pseudovariety of locally trivial semigroups.