  Let $G$ be a compact connected Lie group and $P \to M$ a smooth principal
$G$-bundle. Let a `cylinder function' on the space $\A$ of smooth connections
on $P$ be a continuous function of the holonomies of $A$ along finitely many
piecewise smoothly immersed curves in $M$, and let a generalized measure on
$\A$ be a bounded linear functional on cylinder functions. We construct a
generalized measure on the space of connections that extends the uniform
measure of Ashtekar, Lewandowski and Baez to the smooth case, and prove it is
invariant under all automorphisms of $P$, not necessarily the identity on the
base space $M$. Using `spin networks' we construct explicit functions spanning
the corresponding Hilbert space $L^2(\A/\G)$, where $\G$ is the group of gauge
transformations.
