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We present a new upper limit on the cosmic molecular gas density at z=2.4−3.4 obtained using the first year of observations from the CO Mapping Array Project (COMAP). COMAP data cubes are stacked on the 3D positions of 243 quasars selected from the Extended Baryon Oscillation Spectroscopic Survey (eBOSS) catalog, yielding a 95% upper limit for flux from CO(1-0) line emission of 0.129 Jy km/s. Depending on the balance of the emission between the quasar host and its environment, this value can be interpreted as an average CO line luminosity L′CO of eBOSS quasars of ≤1.26×1011 K km pc2 s−1, or an average molecular gas density ρH2 in regions of the universe containing a quasar of ≤1.52×108 M⊙ cMpc−3. The L′CO upper limit falls among CO line luminosities obtained from individually-targeted quasars in the COMAP redshift range, and the ρH2 value is comparable to upper limits obtained from other Line Intensity Mapping (LIM) surveys and their joint analyses. Further, we forecast the values obtainable with the COMAP/eBOSS stack after the full 5-year COMAP Pathfinder survey. We predict that a detection is probable with this method, depending on the CO properties of the quasar sample. Based on the achieved sensitivity, we believe that this technique of stacking LIM data on the positions of traditional galaxy or quasar catalogs is extremely promising, both as a technique for investigating large galaxy catalogs efficiently at high redshift and as a technique for bolstering the sensitivity of LIM experiments, even with a fraction of their total expected survey data.