Experiments have lagged theory in exploring chemical interactions at temperatures so low that translational degrees of freedom can no longer be treated classically. The difficulty has been to realize in the laboratory low-enough collisional velocities between neutral reactants to access this regime. We report here the realization of merged neutral supersonic beams and the manifestation of clear nonclassical effects in the resulting reactions. We observed orbiting resonances in the Penning ionization reaction of argon and molecular hydrogen with metastable helium, leading to a sharp absolute ionization rate increase in the energy range corresponding to a few degrees kelvin down to 10 millikelvin. Our method should be widely applicable to many canonical chemical reactions.