  (Abridged) We carried out a comprehensive far-ultraviolet (UV) survey of
^12CO and H_2 column densities along diffuse molecular Galactic sight lines in
order to explore in detail the relationship between CO and H_2. We measured new
CO abundances from HST spectra, new H_2 abundances from FUSE data, and new CH,
CH^+, and CN abundances from the McDonald and European Southern Observatories.
A plot of log N(CO) versus log N(H_2) shows that two power-law relationships
are needed for a good fit of the entire sample, with a break located at log
N(CO, cm^-2) = 14.1 and log N(H_2) = 20.4, corresponding to a change in
production route for CO in higher-density gas. Similar logarithmic plots among
all five diatomic molecules allow us to probe their relationships, revealing
additional examples of dual slopes in the cases of CO versus CH (break at log N
= 14.1, 13.0), CH^+ versus H_2 (13.1, 20.3), and CH^+ versus CO (13.2, 14.1).
These breaks are all in excellent agreement with each other, confirming the
break in the CO versus H_2 relationship, as well as the one-to-one
correspondence between CH and H_2 abundances. Our new sight lines were selected
according to detectable amounts of CO in their spectra and they provide
information on both lower-density (< 100 cm^-3) and higher-density diffuse
clouds. The CO versus H_2 correlation and its intrinsic width are shown to be
empirically related to the changing total gas density among the sight lines of
the sample. We employ both analytical and numerical chemical schemes in order
to derive details of the molecular environments. In the low-density gas, where
equilibrium-chemistry studies have failed to reproduce the abundance of CH^+,
our numerical analysis shows that nonequilibrium chemistry must be employed for
correctly predicting the abundances of both CH^+ and CO.
