  X-ray emission is one of the signposts of circumstellar interaction in
supernovae (SNe), but until now, it has been observed only in core-collapse
SNe. The level of thermal X-ray emission is a direct measure of the density of
the circumstellar medium (CSM), and the absence of X-ray emission from Type Ia
SNe has been interpreted as a sign of a very low density CSM. In this paper, we
report late-time (500--800 days after discovery) X-ray detections of SN 2012ca
in {\it Chandra} data. The presence of hydrogen in the initial spectrum led to
a classification of Type Ia-CSM, ostensibly making it the first SN~Ia detected
with X-rays. Our analysis of the X-ray data favors an asymmetric medium, with a
high-density component which supplies the X-ray emission. The data suggest a
number density $> 10^8$ cm$^{-3}$ in the higher-density medium, which is
consistent with the large observed Balmer decrement if it arises from
collisional excitation. This is high compared to most core-collapse SNe, but it
may be consistent with densities suggested for some Type IIn or superluminous
SNe. If SN 2012ca is a thermonuclear SN, the large CSM density could imply
clumps in the wind, or a dense torus or disk, consistent with the
single-degenerate channel. A remote possibility for a core-degenerate channel
involves a white dwarf merging with the degenerate core of an asymptotic giant
branch star shortly before the explosion, leading to a common envelope around
the SN.
