It has recently become possible to focus a beam of high-energy electrons to a spot smaller than an atom, with vast potential for the study of condensed matter. We devise an experiment that can image, with subangstrom resolution, the distribution of such an electron beam as it scatters within an atomic lattice. Our experiments reveal the acute sensitivity of the scattered electron distribution to 0.1 Å shifts of the impact point. Scattering due to plasmon excitations is also examined. Implications for the study of the atomic and electronic structure of condensed matter are discussed.