High-resolution X-ray diffraction data, coupled with theoretical calculations, are used to demonstrate the presence of a non-nuclear local maximum in the electron density of a dimeric Mg(I) molecule. This is the first time such a non-nuclear attractor (NNA) has been observed in a stable molecular species. Multipole modeling of the Mg(I) centers requires use of expansion/contraction (κ) coefficients taken from density functional theory (DFT), since accurate scattering factors for Mg(I) are not available. The model developed accurately accounts for the electron density in the Mg-Mg region and is in excellent agreement with directly calculated DFT data. Within the quantum theory of atoms in molecules (QTAIM), this molecule is not bound by a Mg-Mg bond but rather by two Mg-"pseudo-atom" bonds. The NNA is associated with a large region of negative Laplacian in the Mg-Mg internuclear region and arises from the overlap of 3s orbitals in this long, nonpolar "bond". The pseudoatomic basin associated with the NNA contains 0.8 electrons, which are highly delocalized and hence weakly bound. Possible implications of this unusual electronic structure for the chemistry of such molecules, including their use as excellent reducing agents, are discussed.