Dual-phase xenon detectors lead the search for keV-scale nuclear recoil signals expected from the scattering of weakly interacting massive particle (WIMP) dark matter, and can potentially be used to study the coherent nuclear scattering of MeV-scale neutrinos. New capabilities of such experiments can be enabled by extending their nuclear recoil searches down to the lowest measurable energy. The response of the liquid xenon target medium to nuclear recoils, however, is not well characterized below a few keV, leading to large uncertainties in projected sensitivities. In this work, we report a new measurement of ionization signals from nuclear recoils in liquid xenon down to the lowest energy reported to date. At 0.3 keV, we find that the average recoil produces approximately one ionization electron; this is the first measurement of nuclear recoil signals at the single-ionization-electron level, approaching the physical limit of liquid xenon ionization detectors. We discuss the implications of these measurements on the physics reach of xenon detectors for nuclear-recoil-based WIMP dark matter searches and the detection of coherent elastic neutrino-nucleus scattering.