  We report a photometric and spectroscopic study of the peculiar cataclysmic
variable SDSS 1507+52. The star shows very deep eclipses on the 67 minute
orbital period, and those eclipses are easily separable into white-dwarf and
hot-spot components. This leads to tight constraints on binary parameters, with
M_1=0.83(8) solar masses, M_2=0.057(8) solar masses, R_1=0.0097(9) solar radii,
R_2=0.097(4) solar radii, q=0.069(2), and i=83.18(13) degrees. Such numbers
suggest possible membership among the WZ Sge stars, a common type of dwarf
nova. The spectroscopic behavior (strong and broad H emission, double-peaked
and showing a classic rotational disturbance during eclipse) is also typical.
But the star's orbital period is shockingly below the "period minimum" of
approximately 77 min which is characteristic of hydrogen-rich CVs; producing
such a strange binary will require some tinkering with the theory of
cataclysmic-variable evolution. The proper motion is also remarkably high for a
star of its distance, which we estimate from photometry and trigonometric
parallax as 230 +- 40 pc. This suggests a transverse velocity of 164 +- 30 km/s
- uncomfortably high if the star belongs to a Galactic-disk population. These
difficulties with understanding its evolution and space velocity can be solved
if the star belongs to a Galactic-halo population.
