We measure electronic conductance through single conjugated molecules bonded to Au metal electrodes with direct Au-C covalent bonds using the scanning tunneling microscope based break-junction technique. We start with molecules terminated with trimethyltin end groups that cleave off in situ, resulting in formation of a direct covalent σ bond between the carbon backbone and the gold metal electrodes. The molecular carbon backbone used in this study consist of a conjugated π system that has one terminal methylene group on each end, which bonds to the electrodes, achieving large electronic coupling of the electrodes to the π system. The junctions formed with the prototypical example of 1,4-dimethylenebenzene show a conductance approaching one conductance quantum (G(0) = 2e(2)/h). Junctions formed with methylene-terminated oligophenyls with two to four phenyl units show a 100-fold increase in conductance compared with junctions formed with amine-linked oligophenyls. The conduction mechanism for these longer oligophenyls is tunneling, as they exhibit an exponential dependence of conductance on oligomer length. In addition, density functional theory based calculations for the Au-xylylene-Au junction show near-resonant transmission, with a crossover to tunneling for the longer oligomers.