  A Weakly Interacting Massive Particle (WIMP) provides an attractive dark
matter candidate, and should be within reach of the next generation of
high-energy colliders. We consider the process of direct WIMP pair-production,
accompanied by an initial-state radiation photon, in electron-positron
collisions at the proposed International Linear Collider (ILC). We present a
parametrization of the differential cross section for this process which
conveniently separates the model-independent information provided by cosmology
from the model-dependent inputs from particle physics. As an application, we
consider two simple models, one supersymmetric, and another of the "universal
extra dimensions" (UED) type. The discovery reach of the ILC and the expected
precision of parameter measurements are studied in each model. In addition, for
each of the two examples, we also investigate the ability of the ILC to
distinguish between the two models through a shape-discrimination analysis of
the photon energy spectrum. We show that with sufficient beam polarization the
alternative model interpretation can be ruled out in a large part of the
relevant parameter space.
