We have measured directly the residual energy of cosmic ray muons crossing the MACRO detector at the Gran Sasso Laboratory. For this measurement we have used a transition radiation detector consisting of three identical modules, each of about 12m2 area, operating in the energy region from 100 GeV to 1 TeV. The results presented here were obtained with the first module collecting data for more than two years. The average single muon energy is found to be 320 ± 4 (stat.) ± 11 (syst.) GeV in the rock depth range 3000-6500 hg/cm2. The results are in agreement with calculations of the energy loss of muons in the rock above the detector.