Much of the work in loop quantum gravity and quantum geometry rests on a
mathematically rigorous integration theory on spaces of distributional
connections. Most notably, a diffeomorphism invariant representation of the
algebra of basic observables of the theory, the Ashtekar-Isham-Lewandowski
representation, has been constructed. Recently, several uniqueness results for
this representation have been worked out. In the present article, we contribute
to these efforts by showing that the AIL-representation is irreducible,
provided it is viewed as the representation of a certain C*-algebra which is
very similar to the Weyl algebra used in the canonical quantization of free
quantum field theories.