The structure of ({radical}3 {times} {radical}3 ) Ag/Si(111) is investigated using first-principles total-energy calculations. The lowest-energy configuration consists of a top layer of Ag atoms arranged as honeycomb chained trimers'' lying above a distorted missing top layer'' Si(111) substrate. We find that the honeycomb structure observed in scanning-tunneling-microscope images arises from the electronic charge density on an empty surface band near the Fermi level.