We study relatively invariant measures with the multiplicators
Q
+
∗
∋
q
↦
q
−
β
{\mathbb Q}^*_+\ni q\mapsto q^{-\beta }
on the space
A
f
\mathcal A_f
of finite adeles. We prove that for
β
∈
(
0
,
1
]
\beta \in (0,1]
such measures are ergodic, and then deduce from this the uniqueness of KMS
β
_\beta
-states for the Bost-Connes system. Combining this with a result of Blackadar and Boca-Zaharescu, we also obtain ergodicity of the action of
Q
∗
\mathbb Q^*
on the full adeles.