  In this study, the sensitivity of future lepton colliders to WIMP dark matter
is evaluated assuming WIMP pair production accompanied by a photon from initial
state radiation, through which the process can be identified. A full detector
simulation for the International Large Detector (ILD) concept at the
International Linear Collider (ILC) is performed for a \com energy of 500\,GeV.
Energy scales of up to $3$\,TeV can be tested for different effective operators
for WIMP masses almost up to half the \com energy. The sensitivity benefits
from the polarised beams, which can reduce the main SM background from neutrino
pair production substantially. In addition, systematic uncertainties are shown
to be significantly reduced by combining data with several different
polarisation configurations. In comparison to a previous study, the
reconstruction of the forward detectors has been improved and the systematic
uncertainties are fully treated. The results are also extrapolated to other
\com energies, luminosities and beam polarisations. This allows to provide
results for the full ILC programme, i.e.\ from 250\,GeV to 1\,TeV, as well as
to give approximate results for other planned lepton colliders.
