  Noncommutative U(1) gauge theory on the Moyal-Weyl space ${\bf
R}^2{\times}{\bf R}^2_{\theta}$ is regularized by approximating the
noncommutative spatial slice ${\bf R}^2_{\theta}$ by a fuzzy sphere of matrix
size $L$ and radius $R$ . Classically we observe that the field theory on the
fuzzy space ${\bf R}^2{\times}{\bf S}^2_L$ reduces to the field theory on the
Moyal-Weyl plane ${\bf R}^2{\times}{\bf R}^2_{\theta}$ in the flattening
continuum planar limits $R,L{\longrightarrow}{\infty}$ where
$R^2/L^{2q}{\simeq}{\theta}^2/4^q$ and $q>{3/2}$ . The effective
noncommutativity parameter is found to be given by
${\theta}_{eff}^2{\sim}2{\theta}^2(\frac{L}{2})^{2q-1}$ and thus it corresponds
to a strongly noncommuting space. In the quantum theory it turns out that this
prescription is also equivalent to a dimensional reduction of the model where
the noncommutative U(1) gauge theory in 4 dimensions is shown to be equivalent
in the large $L$ limit to an ordinary $O(M)$ non-linear sigma model in 2
dimensions where $M{\sim}3L^2$ . The Moyal-Weyl model defined this way is also
seen to be an ordinary renormalizable theory which can be solved exactly using
the method of steepest descents . More precisely we find for a fixed
renormalization scale $\mu$ and a fixed renormalized coupling constant $g_r^2$
an $O(M)-$symmetric mass, for the different components of the sigma field,
which is non-zero for all values of $g_r^2$ and hence the $O(M)$ symmetry is
never broken in this solution . We obtain also an exact representation of the
beta function of the theory which agrees with the known one-loop perturbative
result .
