Particle detectors that use liquid xenon (LXe) as detection medium are among the leading technologies in the search for dark matter weakly interacting massive particles (WIMPs). A key enabling element has been the low-energy detection threshold for recoiling nuclei produced by the interaction of WIMPs in LXe targets. In these detectors, the nuclear recoil energy scale is based on the LXe scintillation signal and thus requires knowledge of the relative scintillation efficiency of nuclear recoils, L eff . The uncertainty in L eff at low energies is the largest systematic uncertainty in the reported results from LXe WIMP searches at low masses. In the context of the XENON Dark Matter project, a new LXe scintillation detector has been designed and built specifically for the measurement of L eff at low energies, with an emphasis on maximizing the scintillation light detection efficiency to obtain the lowest possible energy threshold. We report new measurements of L eff at low energies performed with this detector. Our results suggest a L eff which slowly decreases with decreasing energy, from 0.144 ± 0.009 at 15 keV down to 0.088 +0.014 −0.015 at 3 keV.