  We present measurements of the emission from the centers of fifteen spiral
galaxies in the 3P1-3P0 [CI] fine-structure transition at 492 GHz. Observed
galaxy centers range from quiescent to starburst to active. The intensities of
neutral carbon, the J=2-1 transition of 13CO and the J=4-3 transition of 12CO
are compared in matched beams. Most galaxy centers emit more strongly in [CI]
than in 13CO, completely unlike the situation pertaining to Galactic molecular
cloud regions. [CI] intensities are lower than, but nevertheless comparable to
J=4-3 12CO intensities, again rather different from Galactic sources. The ratio
of [CI] to 13CO increases with the central [CI] luminosity of a galaxy; it is
lowest for quiescent and mild starburst centers, and highest for strong
starburst centers and active nuclei. Comparison with radiative transfer model
calculations shows that most observed galaxy centers have neutral carbon
abundances close to, or exceeding, carbon monoxide abundances, rather
independent from the assumed model gas parameters. The same models suggest that
the emission from neutral carbon and carbon monoxide, if assumed to originate
in the same volumes, arises from a warm and dense gas rather than a hot and
tenuous, or a cold and very dense gas. The observed [CI] intensities together
with literature [CII] line and far-infrared continuum data likewise suggest
that a significant fraction of the emission originates in medium-density gas (n
= 10**3 - 10**4 cm**-3, subjected to radiation fields of various strengths.
