The positional order of dodecanethiol monolayers self-assembled on Au(111) is investigated with scanning-tunneling microscopy using ultrahigh tunneling resistances R(t) greater than or equal to 300 G Omega. We have studied two kinds of monolayers prepared (a) by immersion of the Au substrate into a methanoic thiol solution (as-adsorbed films), and (b) by an additional ex situ heat treatment (annealed films). The head-group of the molecules are found to chemisorb in the commensurate root 3 x root 3 overlayer of the Au(111) surface. As-adsorbed monolayers can be characterized by an assembly of nearest-neighbor (nn) rows of molecules with many missing nn rows. Annealed films, on the other hand, are characterized by a 2 x 4 superstructure and zig-zag shaped rows of missing chemisorbed molecules. The two distinct structures are proposed to originate from two different lattices of the carbon-backbone orientational degree of freedom. Annealed films show, in addition, a considerable reduction of defect structures. Annealing to greater than or similar to 100 degrees C results in the desorption of a fraction of the molecules; the remaining molecules stay positionally ordered and are arranged in a "striped domain" pattern.