  Aims: We wish to separate and quantify the CO luminosity and CO-H2 conversion
factor applicable to diffuse but partially-molecular ISM when H2 and CO are
present but C+ is the dominant form of gas-phase carbon.
  Methods: We discuss galactic lines of sight observed in \HI, HCO+ and CO
where CO emission is present but the intervening clouds are diffuse (locally
\AV\ $\la 1$ mag) with relatively small CO column densities $\NCO \la
2\times10^{16}\pcc$. We separate the atomic and molecular fractions
statistically using \EBV\ as a gauge of the total gas column density and
compare NH2 to the observed CO brightness.
  Results: Although there are H2-bearing regions where CO emission is too faint
to be detected, the mean ratio of integrated CO brightness to NH2 for diffuse
ISM does not differ from the usual value of 1\K km/s of integrated CO
brightness per $2\times10^{20}$ H2 $\pcc$ . Moreover, the luminosity of diffuse
CO viewed perpendicular to the galactic plane is 2/3 that seen at the Solar
galactic radius in surveys of CO emission near the galactic plane.
  Conclusions: Commonality of the CO-H2 conversion factors in diffuse and dark
clouds can be understood from considerations of radiative transfer and CO
chemistry. There is unavoidable confusion between CO emission from diffuse and
dark gas and misattribution of CO emission from diffuse to dark or giant
molecular clouds. The character of the ISM is different from what has been
believed if CO and H2 that have been attributed to molecular clouds on the
verge of star formation are actually in more tenuous, gravitationally-unbound
diffuse gas.
