  In this paper we introduce the Integration Problem for principal connections.
Just as a principal connection on a principal bundle $\phi:Q\rightarrow M$ may
be used to split $TQ$ into horizontal and vertical subbundles, a discrete
connection may be used to split $Q\times Q$ into horizontal and vertical
submanifolds. All discrete connections induce a connection on the same
principal bundle via a process known as the Lie or derivative functor. The
Integration Problem consists of describing, for a principal connection
$\mathcal{A}$, the set of all discrete connections whose associated connection
is $\mathcal{A}$. Our first result is that for \emph{flat} principal
connections, the Integration Problem has a unique solution among the
\emph{flat} discrete connections. More broadly, under a fairly mild condition
on the structure group $G$ of the principal bundle $\phi$, we prove that the
existence part of the Integration Problem has a solution that needs not be
unique. Last, we see that, when $G$ is abelian, given compatible continuous and
discrete curvatures the Integration Problem has a unique solution constrained
by those curvatures.
