  With a spin frequency of 707 Hz, PSR J0952-0607 is the second fastest
spinning pulsar known. It was discovered in radio by LOFAR in 2017 at an
estimated distance of either 0.97 or 1.74 kpc and has a low-mass companion with
a 6.42 hr orbital period. We report discovery of the X-ray counterpart of PSR
J0952-0607 using XMM-Newton. The X-ray spectra can be well-fit by a single
power law model (Gamma = 2.5) or by a thermal plus power law model (kTeff = 40
eV and Gamma = 1.4). We do not detect evidence of variability, such as that due
to orbital modulation from pulsar wind and companion star interaction. Because
of its fast spin rate, PSR J0952-0607 is a crucial source for understanding the
r-mode instability, which can be an effective mechanism for producing
gravitational waves. Using the high end of our measured surface temperature, we
infer a neutron star core temperature of ~10^7 K, which places PSR J0952-0607
within the window for the r-mode to be unstable unless an effect such as
superfluid mutual friction damps the fluid oscillation. The measured luminosity
limits the dimensionless r-mode amplitude to be less than ~1x10^-9.
