Plasmonic sensors are extremely promising candidates for label-free single-molecule analysis but require exquisite control over the physical arrangement of metallic nanostructures. Here we employ self-assembly based on the DNA origami technique for accurate positioning of individual gold nanoparticles. Our innovative design leads to strong plasmonic coupling between two 40 nm gold nanoparticles reproducibly held with gaps of 3.3 +/- 1 nm. This is confirmed through far field scattering measurements on individual dimers which reveal a significant red shift in the plasmonic resonance peaks, consistent with the high dielectric environment due to the surrounding DNA. We use surface-enhanced Raman scattering (SERS) to demonstrate local field enhancements of several orders of magnitude through detection of a small number of dye molecules as well as short single-stranded DNA oligonucleotides. This demonstrates that DNA origami is a powerful tool for the high-yield creation of SERS-active nanoparticle assemblies with reliable sub-5 nm gap sizes.