  The extended ultraviolet (XUV) disk galaxies are one of the most interesting
objects studied in the last few years. The UV emission, revealed by GALEX,
extends well beyond the optical disk, after the drop of H$\alpha$ emission, the
usual tracer of star formation. This shows that sporadic star formation can
occur in a large fraction of the HI disk, at radii up to 3 or 4 times the
optical radius. In most galaxies, these regions are poor in stars and dominated
by under-recycled gas, therefore bear some similarity to early stages of spiral
galaxies and high-redshift galaxies.
  One remarkable example is M83, a nearby galaxy with an extended UV disk
reaching 2 times the optical radius. It offers the opportunity to search for
the molecular gas and characterise the star formation in outer disk regions,
traced by the UV emission. We obtained CO(2-1) observations with
  ALMA of a small region in a 1.5'$\times$ 3' rectangle located at
$r_{gal}=7.85'$ over a bright UV region of M83. There is no CO detection, in
spite of the abundance of HI gas, and the presence of young stars traced by
their HII regions. Our spatial resolution (17pc x 13pc) was perfectly fitted to
detect Giant Molecular Clouds (GMC), but none were detected. The corresponding
upper limits occur in an SFR region of the Kennicutt-Schmidt diagram where
dense molecular clouds are expected. Stacking our data over HI-rich regions,
using the observed HI velocity, we obtain a tentative detection, corresponding
to an H$_2$-to-HI mass ratio of $<$ 3 $\times$ 10$^{-2}$. A possible
explanation is that the expected molecular clouds are CO-dark, because of the
strong UV radiation field. The latter preferentially dissociates CO with
respect to H$_2$, due to the small size of the star forming clumps in the outer
regions of galaxies.
