We report the electrical conductance at the single molecule level of the oligoyne molecular wires Py-(C[triple bond]C)(n)-Py (n = 1, 2 and 4; Py = 4-pyridyl) using STM-molecular break junction techniques in Au|molecule|Au configurations. The conductance histograms reveal multiple series of peaks attributed to differing contact geometries between the pyridyl head groups and the gold electrodes. Both experimental and theoretical evidence point to the higher conduction groups being related to adsorption of the pyridyl group at more highly coordinated sites such as step edges or alongside gold adatoms. All three conduction groups in the oligoyne series show a remarkably low beta value of (0.06 +/- 0.03) A(-1), that is, the conductance is almost independent of molecular length. 4,4'-Bipyridyl studied under the same conditions does not follow this exponential decay series. Theoretical calculations using a combination of density functional theory and nonequilibrium Green's function formalism support the experimental results. We conclude that oligoynes and polyynes are a very promising class of molecular wires for integration into electronic circuitry.