A class of materials known as "ferroelectric metals" was discussed theoretically by Anderson and Blount in 1965 [Phys. Rev. Lett. 14, 217 (1965)]], but to date no examples of this class of materials have been reported. Here we present measurements of the elastic moduli of Cd2Re2O7 through the 200 K cubic-to-tetragonal phase transition. A Landau analysis of the moduli reveals that the transition is consistent with Cd2Re2O7 being classified as a ferroelectric metal in the weaker sense described by Anderson and Blount (loss of a center of symmetry). First-principles calculations of the lattice instabilities indicate that the dominant lattice instability corresponds to a twofold degenerate mode with E(u) symmetry and that motions of the O ions forming the O octahedra dominate the energetics of the transition.