The mass of the Z boson has been determined by combining the data from the four LEP experiments ALEPH, DELPHI, L3 and OPAL. The dominant error arises from uncertainties in the calibration of the energy of the beams in LEP. A programme of investigations including energy calibration by resonant depolarization of transversely polarized beams has led to a significant reduction of the uncertainty on the Z mass compared with the precision achieved with the 1990 data. The mass of the Z is measured to be M(Z) = (91.187 +/- 0.007) GeV.