A study of the scaling properties of the electron energy spectrum for a one-dimensional tight-binding model with Thue-Morse on-site potential is presented. This system is interesting because it is not periodic, quasiperiodic or random. The multifractal scaling functions f(α) and τ(q) have been numerically calculated using periodic approximations of the spectrum. In the thermodynamic limit, it appears that τ(q) = q − 1 for q < 1 and τ(q) = 0 for q > 1; correspondingly, the f(α) curve reduces to the two points (α=1, f=1) and (α=1, f=0). This behavior is not sensitive to the strenght of the Thue-Morse potential in the range investigated and it is quite different from the behavior exhibited by quasiperiodic systems. The present results suggest that the spectrum of the Thue-Morse lattice contains absolutely continuous parts (band-like extended states) plus pure point singularities (atomic-like localized states).