An open challenge for single-molecule electronics is to find stable contacts at room temperature with a well-defined conductance. Common coinage metal electrodes pose fabrication and operational problems due to the high mobility of the surface atoms. We demonstrate how molecules covalently grafted onto mechanically robust graphite/graphene substrates overcome these limitations. To this aim, we explore the effect of the anchoring group chemistry on the charge transport properties of graphite-molecule contacts by means of the scanning tunneling microscopy break-junction technique and ab initio simulations. Molecules adsorbed on graphite only via van der Waals interactions have a conductance that decreases exponentially upon stretching the junctions, whereas the molecules bonded covalently to graphite have a single well-defined conductance and yield contacts of unprecedented stability at room temperature. Our results demonstrate a strong bias dependence of the single-molecule conductance, which varies over more than one order of magnitude even at low bias voltages, and show an opposite rectification behavior for covalent and noncovalent contacts. We demonstrate that this bias-dependent conductance and opposite rectification behavior is due to a novel effect caused by the nonconstant, highly dispersive density of states of graphite around the Fermi energy and that the direction of rectification is governed by the detailed nature of the molecule/graphite contact. Combined with the prospect of new functionalities due to a strongly bias-dependent conductance, these covalent contacts are ideal candidates for next-generation molecular electronic devices.