  We introduce a $\mathbb{C}/\mathbb{Z}$-valued invariant of a foliated
manifold with a stable framing and with a partially flat vector bundle. This
invariant can be expressed in terms of integration in differential $K$-theory,
or alternatively, in terms of $\eta$-invariants of Dirac operators and local
correction terms. Initially, the construction of the element in
$\mathbb{C}/\mathbb{Z}$ involves additional choices. But if the codimension of
the foliation is sufficiently small, then this element is independent of these
choices and therefore an invariant of the data listed above. We show that the
invariant comprises various classical invariants like Adams' $e$-invariant, the
$\rho$-invariant of twisted Dirac operators, or the Godbillon-Vey invariant
from foliation theory. Using methods from differential cohomology theory we
construct a regulator map from the algebraic $K$-theory of smooth functions on
a manifold to its connective $K$-theory with $\mathbb{C}/\mathbb{Z}$
coefficients. Our main result is a formula for the invariant in terms of this
regulator and integration in algebraic and topological $K$-theory.
