  Using a new scaling limit as well as a new cut-off procedure, we show that
$\phi^4$ theory on noncommutative ${\bf R}^4$ can be obtained from the
corresponding theory on fuzzy ${\bf S}^2 \times {\bf S}^2$. The star-product on
this noncommutative ${\bf R}^4$ is effectively local in the sense that the
theory naturally has an ultra-violet cut-off $\Lambda$ which is inversely
proportional to the noncommutativity $\theta$, i.e $ \Lambda=
\frac{2}{\theta}$. We show that the UV-IR mixing in this case is absent to one
loop in the $2-$point function and also comment on the $4-$point function.
