Non-equilibrium Green's functions calculations based on density functional theory show a direct link between the initial stages of H$_2$ dissociation on a gold atomic wire and the electronic current supported by the gold wire. The simulations reveal that for biases below the stability threshold of the wire, the minimum-energy path for H$_2$ dissociation is not affected. However, the electronic current presents a dramatic drop when the molecule initiates its dissociation. This current drop is traced back to quantum interference between electron paths when the molecule starts interacting with the gold wire.