ABSTRACT We present a pilot search of CO emission in three H2-absorbing, long-duration gamma-ray burst (GRB) host galaxies at z ∼ 2–3. We used the Atacama Large Millimeter/submillimeter Array (ALMA) to target the CO(3 − 2) emission line and report non-detections for all three hosts. These are used to place limits on the host molecular gas masses, assuming a metallicity-dependent CO-to-H2 conversion factor (αCO). We find, $M_{\rm mol} \lt 3.5\times 10^{10}\, M_{\odot }$ (GRB 080607), $M_{\rm mol} \lt 4.7\times 10^{11}\, M_{\odot }$ (GRB 120815A), and $M_{\rm mol} \lt 8.9\times 10^{11}\, M_{\odot }$ (GRB 181020A). The high limits on the molecular gas mass for the latter two cases are a consequence of their low stellar masses M⋆ ($M_\star \lesssim 10^{8}\, M_{\odot }$) and low gas-phase metallicities ($Z\sim 0.03\, Z_{\odot }$). The limit on the Mmol/M⋆ ratio derived for GRB 080607, however, is consistent with the average population of star-forming galaxies at similar redshifts and stellar masses. We discuss the broader implications for a metallicity-dependent CO-to-H2 conversion factor and demonstrate that the canonical Galactic αCO will severely underestimate the actual molecular gas mass for all galaxies at z > 1 with $M_\star \lt 10^{10}\, M_\odot$. To better quantify this we develop a simple approach to estimate the relevant αCO factor based only on the redshift and stellar mass of individual galaxies. The elevated conversion factors will make these galaxies appear CO-‘dark’ and difficult to detect in emission, as is the case for the majority of GRB hosts. GRB spectroscopy thus offers a complementary approach to identify low-metallicity, star-forming galaxies with abundant molecular gas reservoirs at high redshifts that are otherwise missed by current ALMA surveys.