In molecular electronics individual molecules serve as electronic devices. In these systems, electron-vibron (e-nu) coupling can be expected to lead to new physical phenomena and potential device functions(1-3). In previous studies of molecular wires, the e-nu coupling occurred as a result of the well-known Franck-Condon principle, for which the Born-Oppenheimer approximation holds. This means that after a vibronic excitation, the electrons and the vibrations evolve independently from each other. Here we show that this simple picture changes markedly when two electronic levels in a molecule are coupled by a molecular vibration(4,5). In molecular wires we observe a non-Born-Oppenheimer regime, for which a coherent coupling of electronic and nuclear motion emerges(6). This phenomenon should occur in all systems with strong electron-vibration coupling and an electronic level spacing of the order of vibrational energies. The coherent coupling of electronic and nuclear motion could be used to implement mechanical control of electron transport in molecular electronics.