In the presence of matter, there is no fundamental limit preventing confinement of visible light even down to atomic scales. Achieving such confinement and the corresponding resonant intensity enhancement inevitably requires simultaneous control over atomic-scale details of material structures and over the optical modes that such structures support. By means of self-assembly we have obtained side-by-side aligned gold nanorod dimers with robust atomically defined gaps reaching below 0.5 nm. The existence of atomically confined light fields in these gaps is demonstrated by observing extreme Coulomb splitting of corresponding symmetric and antisymmetric dimer eigenmodes of more than 800 meV in white-light scattering experiments. Our results open new perspectives for atomically resolved spectroscopic imaging, deeply nonlinear optics, ultrasensing, cavity optomechanics, as well as for the realization of novel quantum-optical devices.