  We complete the classification of Bost--Connes systems. We show that two
Bost--Connes C*-algebras for number fields are isomorphic if and only if the
original semigroups actions are conjugate. Together with recent reconstruction
results in number theory by Cornelissen--de Smit--Li--Marcolli--Smit, we
conclude that two Bost--Connes C*-algebras are isomorphic if and only if the
original number fields are isomorphic.
