  Both the mass (just below the D^{*0}Dbar^0 threshold) and the likely quantum
numbers (J^{PC}=1^{++}) of the X(3872) suggest that it is either a weakly-bound
hadronic ``molecule'' or a virtual state of charmed mesons. Assuming the
X(3872) is a weakly-bound molecule, the scattering of neutral D and D* mesons
off the X(3872) can be predicted from the X(3872) binding energy. We calculate
the phase shifts and cross section for scattering of D^0 and D^{*0} mesons and
their antiparticles off the X(3872) in an effective field theory for
short-range interactions. This provides another example of a three-body
process, along with those in nuclear and atomic systems, that displays
universal properties. It may be possible to extract the scattering within the
final state interactions of B_c decays and/or other LHC events.
