  Using high resolution Chandra data, we report the presence of a weak X-ray
point source coincident with the nucleus of NGC 4178, a late-type bulgeless
disk galaxy known to have high ionization mid-infrared (mid-IR) lines typically
associated with active galactic nuclei (AGNs). Although the faintness of this
source precludes a direct spectral analysis, we are able to infer its basic
spectral properties using hardness ratios. X-ray modeling, combined with the
nuclear mid-IR characteristics, suggests that NGC 4178 may host a highly
absorbed AGN accreting at a high rate with a bolometric luminosity on order of
10^43 ergs/s. The black hole mass estimate, based on our Chandra data and
archival VLA data using the most recent fundamental plane relations is \sim
10^4 - 10^5 M\odot, possibly the lowest mass nuclear black hole currently
known. There are also three off-nuclear sources, two with a similar brightness
to the nuclear source at 36" and 32" from the center. As with the nuclear
source, hardness ratios are used to estimate spectra for these two sources, and
both are consistent with a simple power- law model with absorption. These two
sources have X-ray luminosities of the order of \sim 10^38 ergs/s, which place
them at the threshold between X-ray binaries and ultra-luminous X-ray sources
(ULXs). The third off-nuclear source, located 49" from the center, is the
brightest source detected, with an X-ray luminosity of \sim 10^40 ergs/s. Its
spectrum is well-fit with an absorbed power law model, suggesting that it is a
ULX. We also fit its spectrum with the Bulk Motion Comptonization (BMC) model
and suggest that this source is consistent with an intermediate-mass black hole
(IMBH) of mass (6\times2)\times10^3 M\odot.
