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A planar superconducting undulator (SCU) with a period of 15 mm is under development at the Advanced Photon Source (APS). The SCU is designed to achieve a peak field of 0.8 T on the beam axis for an 8 mm pole gap and a current density in the coil of 1.0 kA/mm2. Short sections of low-carbon-steel cores with 12 and 22 periods were fabricated, and coil windings were completed with NbTi superconducting wire. After "training" by means of quenches, the SCUs were able to charge up to near the critical current density jc of 1.43 kA/mm2. Using a thin-film heater attached to the inner surface of a vacuum chamber wall, steady-state heat fluxes were applied to the coil/pole face of the 12-period SCU in 4.2 K liquid He (LHe). The heat flux densities needed to quench the SCU were measured. At 0.998jc and 0.8jc the thermal stability margins were about 1.3 mW/mm2 and 2 mW/mm2, respectively. The thermal stability margin of the SCU was affected by the latent heat of vaporization of LHe.