Quantum dots are small conductive regions in a semiconductor, containing a variable number of electrons (from one to a thousand) that occupy well-defined, discrete quantum states-for which reason they are often referred to as artificial atoms . Connecting them to current and voltage contacts allows the discrete energy spectra to be probed by charge-transport measurements. Two quantum dots can be connected to form an 'artificial molecule'. Depending on the strength of the inter-dot coupling (which supports quantum-mechanical tunnelling of electrons between the dots), the two dots can form 'ionic' [2-6] or 'covalent' bonds. In the former case, the electrons are localized on individual dots, while in the latter, the electrons are delocalized over both dots. The covalent binding leads to bonding and antibonding states, whose energy difference is proportional to the degree of tunnelling. Here we report a transition from ionic bonding to covalent bonding in a quantum-dot 'artificial molecule' that is probed by microwave excitations [5-8]. Our results demonstrate controllable quantum coherence in single-electron devices, an essential requirement for practical applications of quantum-dot circuitry.