The ZEPLIN-III experiment in the Palmer Underground Laboratory at Boulby uses a 12 kg two-phase xenon time-projection chamber to search for the weakly interacting massive particles (WIMPs) that may account for the dark matter of our Galaxy. The detector measures both scintillation and ionization produced by radiation interacting in the liquid to differentiate between the nuclear recoils expected from WIMPs and the electron-recoil background signals down to $\ensuremath{\sim}10\text{ }\text{ }\mathrm{keV}$ nuclear-recoil energy. An analysis of $847\text{ }\text{ }\mathrm{kg}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{days}$ of data acquired between February 27, 2008, and May 20, 2008, has excluded a WIMP-nucleon elastic scattering spin-independent cross section above $8.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}\text{ }\text{ }\mathrm{pb}$ at $60\text{ }\text{ }\mathrm{GeV}{c}^{\ensuremath{-}2}$ with a 90% confidence limit. It has also demonstrated that the two-phase xenon technique is capable of better discrimination between electron and nuclear recoils at low-energy than previously achieved by other xenon-based experiments.