Fabricating nanocavities in which optically active single quantum emitters are precisely positioned is crucial for building nanophotonic devices. Here we show that self-assembly based on robust DNA-origami constructs can precisely position single molecules laterally within sub-5 nm gaps between plasmonic substrates that support intense optical confinement. By placing single-molecules at the center of a nanocavity, we show modification of the plasmon cavity resonance before and after bleaching the chromophore and obtain enhancements of ≥4 × 103 with high quantum yield (≥50%). By varying the lateral position of the molecule in the gap, we directly map the spatial profile of the local density of optical states with a resolution of ±1.5 nm. Our approach introduces a straightforward noninvasive way to measure and quantify confined optical modes on the nanoscale.