  In this paper we shall investigate the possibility of solving U(1) theories
on the non-commutative (NC) plane for arbitrary values of $\theta$ by
exploiting Morita equivalence. This duality maps the NC U(1) on the two-torus
with a rational parameter $\theta$ to the standard U(N) theory in the presence
of a 't Hooft flux, whose solution is completely known. Thus, assuming a smooth
dependence on $\theta$, we are able to construct a series rational approximants
of the original theory, which is finally reached by taking the large $N-$limit
at fixed 't Hooft flux. As we shall see, this procedure hides some subletities
since the approach of $N$ to infinity is linked to the shrinking of the
commutative two-torus to zero-size. The volume of NC torus instead diverges and
it provides a natural cut-off for some intermediate steps of our computation.
In this limit, we shall compute both the partition function and the correlator
of two Wilson lines. A remarkable fact is that the configurations, providing a
finite action in this limit, are in correspondence with the non-commutative
solitons (fluxons) found independently by Polychronakos and by Gross and
Nekrasov, through a direct computation on the plane.
