We present ^ { 12 } CO 1 \rightarrow 0 observations of eleven low luminosity ( M _ { B } > -18 ) , HI–rich dwarf galaxies .
Only the three most metal-rich galaxies , with 12+log ( O/H ) \approx 8.2 , are detected .
Very deep CO spectra of six extremely metal-poor systems ( 12+log ( O/H ) \leq 7.5 ) yield only low upper limits on the CO surface brightness , I _ { CO } < 0.1 K km s ^ { -1 } .
Three of these six have never before been observed in a CO line , while the others now have much more stringent upper limits .
For the very low metallicity galaxy Leo A , we do not confirm a previously reported detection in CO , and the limits are consistent with another recent nondetection .

We combine these new observations with data from the literature to form a sample of dwarf galaxies which all have CO observations and measured oxygen abundances .
No known galaxies with 12+log ( O/H ) < 7.9 ( Z < 0.1 Z _ { \odot } ) have been detected in CO .
Most of the star-forming galaxies with higher ( 12+log ( O/H ) > 8.1 ) metallicities are detected at similar or higher I _ { CO } surface brightnesses .
The data are consistent with a strong dependence of the I _ { CO } /M _ { H _ { 2 } } ~ { } \equiv~ { } X _ { CO } conversion factor on ambient metallicity .
The strikingly low upper limits on some metal-poor galaxies lead us to predict that the conversion factor is non-linear , increasing sharply below \sim 1/10 of the solar metallicity ( 12+log ( O/H ) \leq 7.9 ) .