Self-assembly of decanethiol on Ag(111) is studied using ultrahigh impedance scanning tunneling microscopy. The surface frequently shows monatomic silver islands covering about 20% of the surface. Molecular domains with an average size of similar to 70 Angstrom as well as the molecules themselves are also imaged. From these images, three important quantities regarding the molecular structure are determined : (1) the lattice constant is 4.61 +/- 0.15 Angstrom; (2) two sets of 6-fold lattice orientations separated by 20.7 +/- 2.3 degrees are observed; (3) far from domain boundaries, height variations of nearest neighbors are less than 0.1 Angstrom. A model is proposed to explain how such a structure could have arisen.