  We present a $C^*$-algebra which is naturally associated to the
$ax+b$-semigroup over $\mathbb N$. It is simple and purely infinite and can be
obtained from the algebra considered by Bost and Connes by adding one unitary
generator which corresponds to addition. Its stabilization can be described as
a crossed product of the algebra of continuous functions, vanishing at
infinity, on the space of finite adeles for $\mathbb Q$ by the natural action
of the $ax+b$-group over $\mathbb Q$.
