  The nearby Chamaeleon molecular cloud complex is a good laboratory to study
the process of low-mass star formation since it consists of three clouds with
very different properties. Cha III does not show any sign of star formation,
while star formation has been very active in Cha I and may already be
finishing. Our goal is to determine whether star formation can proceed in Cha
III, and to compare the results to our recent survey of Cha I. We used the
Large APEX Bolometer Array (LABOCA) to map Cha III in dust continuum emission
at 870 micron. 29 sources are extracted from the map, all of them being
starless. The starless cores are found down to a visual extinction of 1.9 mag,
in marked contrast with other molecular clouds, including Cha I. Apart from
this difference, the Cha III starless cores share very similar properties with
those found in Cha I. At most two sources have a mass larger than the critical
Bonnor-Ebert mass, which suggests that the fraction of prestellar cores is very
low, even lower than in Cha I. Only the most massive sources are candidate
prestellar cores, in agreement with the correlation found earlier in the Pipe
nebula. The mass distribution of the 85 starless cores of Cha I and III that
are not candidate prestellar cores is consistent with a single power law down
to the 90% completeness limit, with an exponent close to the Salpeter value. A
fraction of the starless cores in Cha I and III may still grow in mass and
become gravitationally unstable. Based on predictions of numerical simulations
of turbulent molecular clouds, we estimate that at most 50% and 20% of the
starless cores of Cha I and III, respectively, may form stars. The LABOCA
survey reveals that Cha III, and Cha I to some extent too, is a prime target to
study the formation of prestellar cores, and thus the onset of star formation.
(abridged).
