We present an efficient strategy to modulate tunnelling in molecular junctions by changing the tunnelling decay coefficient, beta, by terminal-atom substitution which avoids altering the molecular backbone. By varying X=H, F, Cl, Br, I in junctions with S(CH2)((10-18))X, current densities (J) increase >4 orders of magnitude, creating molecular conductors via reduction of beta from 0.75 to 0.25 angstrom (-1). Impedance measurements show tripled dielectric constants (epsilon (r)) with X=I, reduced HOMO-LUMO gaps and tunnelling-barrier heights, and 5-times reduced contact resistance. These effects alone cannot explain the large change in beta. Density-functional theory shows highly localized, X-dependent potential drops at the S(CH2)(n)X//electrode interface that modifies the tunnelling barrier shape. Commonly-used tunnelling models neglect localized potential drops and changes in epsilon (r). Here, we demonstrate experimentally that beta 1/<mml:msqrt>epsilon r</mml:msqrt>, suggesting highly-polarizable terminal-atoms act as charge traps and highlighting the need for new charge transport models that account for dielectric effects in molecular tunnelling junctions. In molecular junctions, where a molecule is placed between two electrodes, the current passed decays exponentially as a function of length. Here, Chen et al. show that this exponentially attenuation can be controlled by changing a single atom at the end of the molecular wire.