Context. The nearby Chamaeleon molecular cloud complex is a good laboratory for studying the process of low-mass star formation because it consists of three clouds with very different properties. Chamaeleon III does not show any sign of star formation, while star formation has been very active in Chamaeleon I and may already be finishing. Aims. Our goal is to determine whether star formation can proceed in Cha III by searching for prestellar cores, and to compare the results to our recent survey of Cha I. Methods. We used the Large APEX Bolometer Array (LABOCA) to map Cha III in dust continuum emission at 870 μm. The map is compared with a 2MASS extinction map and decomposed with a multiresolution algorithm. The extracted sources are analyzed by carefully taking into account the spatial filtering inherent in the data reduction process. Results. Twenty-nine sources are extracted from the 870 μm map, all of them starless. The estimated 90% completeness limit is 0.18 M . 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. They are less dense than those detected in continuum emission in other clouds by a factor of a few. At most two sources (<7%) 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 (<17%). 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 detected with LABOCA 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. Conclusions. 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. Obtaining observational constraints on the duration of the core-building phase prior to gravitational collapse will be necessary to make further progress.