XENON10 is an experiment designed to directly detect particle dark matter. It is a dual phase (liquid/gas) xenon time-projection chamber with 3D position imaging. Particle interactions generate a primary scintillation signal (S1) and ionization signal (S2), which are both functions of the deposited recoil energy and the incident particle type. We present a new precision measurement of the relative scintillation yield View the MathML source and the absolute ionization yield View the MathML source, for nuclear recoils in xenon. A dark matter particle is expected to deposit energy by scattering from a xenon nucleus. Knowledge of View the MathML source is therefore crucial for establishing the energy threshold of the experiment; this in turn determines the sensitivity to particle dark matter. Our View the MathML source measurement is in agreement with recent theoretical predictions above 15 keV nuclear recoil energy, and the energy threshold of the measurement is View the MathML source. A knowledge of the ionization yield View the MathML source is necessary to establish the trigger threshold of the experiment. The ionization yield View the MathML source is measured in two ways, both in agreement with previous measurements and with a factor of 10 lower energy threshold.