The main linacs of the proposed International Linear Collider (ILC) consist of superconducting cavities operated at 2 K. The accelerating cavities are contained in a contiguous series of cryogenic modules that also house the main linac quadrupoles, thus the quadrupoles also need to be superconducting. In an early ILC design, these magnets are about 0.6 m long, have cos(2θ) coils, and operate at constant field gradients up to 60 T/m. In order to preserve the small beam emittances in the ILC linacs, the e and e-beams need to traverse the quadrupoles near their magnetic centers. A quadrupole shunting technique is used to measure the quadrupole alignment with the beams; this process requires the magnetic centers move by no more than about 5 micrometers when their strength is changed. To determine if such tight stability is achievable in a superconducting quadrupole, we at SLAC measured the magnetic center motions in a prototype ILC quadrupole built at CIEMAT in Spain. A rotating coil technique was used with a better than 0.1 micrometer precision in the relative field center position, and less than a 2 micrometer systematic error over 30 minutes. This paper describes the warm-bore cryomodule that houses the quadrupole in its Helium vessel, the magnetic center measurement system, the measured center data and strength and harmonics magnetic data. Show Partial Abstract