We present 12 CO 1 → 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)≈ 8.2, are detected. Very deep CO spectra of six extremely metal-poor systems (12+log(O/H)≤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 < 0.1Z ⊙ ) 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 ≡ 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 ∼1/10 of the solar metallicity (12+log(O/H)≤7.9).