  Liquid xenon is an important detection medium in direct dark matter
experiments, which search for low-energy nuclear recoils produced by the
elastic scattering of WIMPs with quarks. The two existing measurements of the
relative scintillation efficiency of nuclear recoils below 20 keV lead to
inconsistent extrapolations at lower energies. This results in a different
energy scale and thus sensitivity reach of liquid xenon dark matter detectors.
We report a new measurement of the relative scintillation efficiency below 10
keV performed with a liquid xenon scintillation detector, optimized for maximum
light collection. Greater than 95% of the interior surface of this detector was
instrumented with photomultiplier tubes, giving a scintillation yield of 19.6
photoelectrons/keV electron equivalent for 122 keV gamma rays. We find that the
relative scintillation efficiency for nuclear recoils of 5 keV is 0.14, staying
constant around this value up to 10 keV. For higher energy recoils we measure a
value around 20%, consistent with previously reported data. In light of this
new measurement, the XENON10 experiment's results on spin-independent
WIMP-nucleon cross section, which were calculated assuming a constant 0.19
relative scintillation efficiency, change from $8.8\times10^{-44}$ cm$^2$ to
$9.9\times10^{-44}$ cm$^2$ for WIMPs of mass 100 GeV/c$^2$, and from
$4.4\times10^{-44}$ cm$^2$ to $5.6\times10^{-44}$ cm$^2$ for WIMPs of mass 30
GeV/c$^2$.
