A detailed study has been made of the feasibility of Møller scattering as a method for continuous monitoring of the LEP beam energy at high energies where the currently used resonant depolarisation technique may be unusable. Two methods, based on Møller scattering, to measure the energy with different dominant systematic errors are discussed. Monte Carlo studies of the effect on the energy measurement of radiative corrections, target binding effects, beam parameters and calorimetric energy resolution have been made. It is shown how the beam parameters necessary to control the systematic error may be deduced from the Møller scattering events, and the crucial target-detector distance monitored using ep scattering events. The beam energy precision is expected to be a few MeV at 90 GeV. The reduction of the systematic error on the W-mass measurement can be appreciable, and for a fixed total error on the W-mass may provide a considerable economy in running time at LEP2. The method is also applicable to precise energy measurement at high-energy electron linear colliders.