The spontaneous formation of organized surface structures at nanometre scales(1,2) has the potential to augment or surpass standard materials patterning technologies. Many observations of self-organization of nanoscale clusters at surfaces have been reported(1-10), but the fundamental mechanisms underlying such behaviour-and in particular, the nature of the forces leading to and stabilizing self-organization-are not well understood. The forces between the many-atom units in these structures, with characteristic dimensions of one to tens of nanometres, must extend far beyond the range of typical interatomic interactions. One commonly accepted source of such mesoscale forces is the stress field in the substrate around each unit(1,11-13). This, however, has not been confirmed, nor have such interactions been measured directly. Here we identify and measure the ordering forces in a nearly perfect triangular lattice of nanometre-sized vacancy islands that forms when a single monolayer of silver on the ruthenium (0001) surface is exposed to sulphur at room temperature. By using time-resolved scanning tunnelling microscopy to monitor the thermal fluctuations of the centres of mass of the vacancy islands around their final positions in the self-organized lattice, we obtain the elastic constants of the lattice and show that the weak forces responsible for its stability can be quantified. Our results are consistent with general theories of strain-mediated interactions between surface defects in strained films.