  Results are presented from the first underground data run of ZEPLIN-II, a 31
kg two phase xenon detector developed to observe nuclear recoils from
hypothetical weakly interacting massive dark matter particles. Discrimination
between nuclear recoils and background electron recoils is afforded by
recording both the scintillation and ionisation signals generated within the
liquid xenon, with the ratio of these signals being different for the two
classes of event. This ratio is calibrated for different incident species using
an AmBe neutron source and Co-60 gamma-ray sources. From our first 31 live days
of running ZEPLIN-II, the total exposure following the application of fiducial
and stability cuts was 225 kgxdays. A background population of radon progeny
events was observed in this run, arising from radon emission in the gas
purification getters, due to radon daughter ion decays on the surfaces of the
walls of the chamber. An acceptance window, defined by the neutron calibration
data, of 50% nuclear recoil acceptance between 5 keVee and 20 keVee, had an
observed count of 29 events, with a summed expectation of 28.6+/-4.3 gamma-ray
and radon progeny induced background events. These figures provide a 90% c.l.
upper limit to the number of nuclear recoils of 10.4 events in this acceptance
window, which converts to a WIMP-nucleon spin-independent cross-section with a
minimum of 6.6x10^-7 pb following the inclusion of an energy dependent,
calibrated, efficiency. A second run is currently underway in which the radon
progeny will be eliminated, thereby removing the background population, with a
projected sensitivity of 2x10^-7 pb for similar exposures as the first run.
