  We introduce a new invariant for C*-algebras of stable rank one that merges
the Cuntz semigroup information together with the K$_1$-group information. This
semigroup, termed the Cu$_1$-semigroup, is constructed as equivalence classes
of pairs consisting of a positive element in the stabilization of the given
C*-algebra together with a unitary element of the unitization of the hereditary
subalgebra generated by the given positive element. We show that the
Cu$_1$-semigroup is a well-defined continuous functor from the category of
C*-algebras of stable rank one to a suitable codomain category that we write
Cu$^\sim$. Furthermore, we compute the Cu$_1$-semigroup of some specific
classes of C*-algebras. Finally, in the course of our investigation, we show
that we can recover functorially Cu, K$_1$ and K$_*:=$K$_0$$\oplus$ K$_1$ from
Cu$_1$.
