  As a first application of our renormalisation group approach to non-local
matrix models [hep-th/0305066], we prove (super-)renormalisability of Euclidean
two-dimensional noncommutative \phi^4-theory. It is widely believed that this
model is renormalisable in momentum space arguing that there would be
logarithmic UV/IR-divergences only. Although momentum space Feynman graphs can
indeed be computed to any loop order, the logarithmic UV/IR-divergence appears
in the renormalised two-point function -- a hint that the renormalisation is
not completed. In particular, it is impossible to define the squared mass as
the value of the two-point function at vanishing momentum. In contrast, in our
matrix approach the renormalised N-point functions are bounded everywhere and
nevertheless rely on adjusting the mass only. We achieve this by introducing
into the cut-off model a translation-invariance breaking regulator which is
scaled to zero with the removal of the cut-off. The naive treatment without
regulator would not lead to a renormalised theory.
