  We present high resolution (~5'') maps of the J = 1 - 0 transitions of
^{13}CO and C^{18}O towards the nucleus of NGC 6946, made with the Owens Valley
Millimeter Array. The images are compared with existing ^{12}CO(1-0) maps to
investigate localized changes in gas properties across the nucleus. As compared
to ^{12}CO, both ^{13}CO and C^{18}O are more confined to the central ring of
molecular gas associated with the nuclear star formation; that is, ^{12}CO is
stronger relative to ^{13}CO and C^{18}O away from the nucleus and along the
spiral arms. The ^{12}CO(1-0)/^{13}CO(1-0) line ratio reaches very high values
of >40. We attribute the relative ^{13}CO weakness to a rapid change in the
interstellar medium from dense star forming cores in a central ring to diffuse,
low density molecular gas in and behind the molecular arms. This change is
abrupt, occurring in less than a beamsize (90 pc), about the size of a giant
molecular cloud. Column densities determined from ^{13}CO(1-0), C^{18}O(1-0),
and 1.4 mm dust continuum all indicate that the standard Galactic conversion
factor, X_{CO}, overestimates the amount of molecular gas in NGC 6946 by
factors of ~3-5 towards the central ring and potentially even more so in the
diffuse gas away from the central starburst. We suggest that the nuclear bar
acts to create coherent regions of molecular clouds with distinct and different
physical conditions. The ^{12}CO(1-0)/^{13}CO(1-0) line ratio in galactic
nuclei can be a signpost of a dynamically evolving ISM.
