The energy loss spectrum of 180 GeV muons has been measured with the 5.6 m long finely segmented Module 0 of the ATLAS hadron Tile Calorimeter at the CERN SPS. The differential probability dP/dv per radiation length of a fractional energy loss v = ΔΕμ/Εμ has been measured in the range 0.025 ≤ v ≤ 0.97; it is compared with theoretical predictions for energy losses due to bremsstrahlung, production of electron-positron pairs, and energetic knock-on electrons. The iron elastic form factor correction ΔelFe = 1.63 ± 0.17stat ± 0.23Syst±0.200.14theor to muon bremsstrahlung in the region of no screening of the nucleus by atomic electrons has been measured for the first time, and is compared with different theoretical predictions.