Weakly Interacting Massive Particles (WIMPs) are one of the leading candidates for Dark Matter. We develop a model-independent method for determining the mass $m_chi$ of the WIMP by using data (i.e., measured recoil energies) of direct detection experiments. Our method is independent of the as yet unknown WIMP density near the Earth, of the form of the WIMP velocity distribution, as well as of the WIMP-nucleus cross section. However, it requires positive signals from at least two detectors with different target nuclei. In a background-free environment, $m_chi sim 50$ GeV could in principle be determined with an error of $sim 35%$ with only $2 times 50$ events; in practice upper and lower limits on the recoil energy of signal events, imposed to reduce backgrounds, can increase the error. The method also loses precision if $m_chi$ significantly exceeds the mass of the heaviest target nucleus used.
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