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).Read less <