  Methods: 12CO emission is imaged in position and position-velocity space
analyzed statistically, and then compared with maps of total reddening and with
models of the C+ - CO transition in H2-bearing diffuse clouds. Results: Around
Zeta Oph, 12CO emission appears in two distinct intervals of reddening centered
near EBV = 0.4 and 0.65 mag, of which < 0.2 mag is background material. Within
either interval, the integrated 12CO intensity varies up to 6-12 K-km/s
compared to 1.5 K-km/s toward Zeta Oph. Nearly 80% of the individual profiles
have velocity dispersions < 0.6 km/s, which are subsonic at the kinetic
temperature derived from H2 toward Zeta Oph, 55 K. Partly as a result, 12CO
emission exposes the internal, turbulent, supersonic (1-3 km/s) gas flows with
especial clarity in the cores of strong lines. The flows are manifested as
resolved velocity gradients in narrow, subsonically-broadened line cores.
Conclusions: The scatter between N(CO) and EBV in global, CO absorption line
surveys toward bright stars is present in the gas seen around Zeta Oph,
reflecting the extreme sensitivity of N(12CO) to ambient conditions. The
two-component nature of the optical absorption toward Zeta Oph is coincidental
and the star is occulted by a single body of gas with a complex internal
structure, not by two distinct clouds. The very bright 12CO lines in diffuse
gas arise at N(H2) ~ 10^21/cm^2 in regions of modest density n(H) ~ 200-500/cc
and somewhat more complete C+-CO conversion. Given the variety of structure in
the foreground gas, it is apparent that only large surveys of absorption
sightlines can hope to capture the intrinsic behavior of diffuse gas.
