The plasmon resonances of two closely spaced metallic particles have enabled applications including single-molecule sensing and spectroscopy, novel nanoantennas, molecular rulers, and nonlinear optical devices. In a classical electrodynamic context, the strength of such dimer plasmon resonances increases monotonically as the particle gap size decreases. In contrast, a quantum mechanical framework predicts that electron tunneling will strongly diminish the dimer plasmon strength for subnanometer-scale separations. Here, we directly observe the plasmon resonances of coupled metallic nanoparticles as their gap size is reduced to atomic dimensions. Using the electron beam of a scanning transmission electron microscope (STEM), we manipulate pairs of ~10-nm-diameter spherical silver nanoparticles on a substrate, controlling their convergence and eventual coalescence into a single nanosphere. We simultaneously employ electron energy-loss spectroscopy (EELS) to observe the dynamic plasmonic properties of these dimers before and after particle contact. As separations are reduced from 7 nm, the dominant dipolar peak exhibits a redshift consistent with classical calculations. However, gaps smaller than ~0.5 nm cause this mode to exhibit a reduced intensity consistent with quantum theories that incorporate electron tunneling. As the particles overlap, the bonding dipolar mode disappears and is replaced by a dipolar charge transfer mode. Our dynamic imaging, manipulation, and spectroscopy of nanostructures enables the first full spectral mapping of dimer plasmon evolution and may provide new avenues for in situ nanoassembly and analysis in the quantum regime.