  Many experiments that aim at the direct detection of Dark Matter are able to
distinguish a dominant background from the expected feeble signals, based on
some measured discrimination parameter. We develop a statistical model for such
experiments using the Profile Likelihood ratio as a test statistic in a
frequentist approach. We take data from calibrations as control measurements
for signal and background, and the method allows the inclusion of data from
Monte Carlo simulations. Systematic detector uncertainties, such as
uncertainties in the energy scale, as well as astrophysical uncertainties, are
included in the model. The statistical model can be used to either set an
exclusion limit or to make a discovery claim, and the results are derived with
a proper treatment of statistical and systematic uncertainties. We apply the
model to the first data release of the XENON100 experiment, which allows to
extract additional information from the data, and place stronger limits on the
spin-independent elastic WIMP-nucleon scattering cross-section. In particular,
we derive a single limit, including all relevant systematic uncertainties, with
a minimum of 2.4x10^-44 cm^2 for WIMPs with a mass of 50 GeV/c^2.
