  We report the discovery of a faint (L_x ~ 4 10^37 erg/s, 0.5-2 keV),
out-flowing gaseous hot interstellar medium (ISM) in NGC 3379. This represents
the lowest X-ray luminosity ever measured from a hot phase of the ISM in a
nearby early type galaxy. The discovery of the hot ISM in a very deep Chandra
observation was possible thanks to its unique spectral and spatial signatures,
which distinguish it from the integrated stellar X-ray emission, responsible
for most of the unresolved emission in the Chandra data. This hot component is
found in a region of about 800 pc in radius at the center of the galaxy and has
a total mass M~ 3 10^5 solar masses. Independent theoretical prediction of the
characteristics of an ISM in this galaxy, based on the intrinsic properti es of
NGC 3379, reproduce well the observed luminosity, temperature, and radial
distribution and mass of the hot gas, and indicate that the gas is in an
outflowing phase, predicted by models but not observed in any system so far.
