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Using the Large Millimeter Telescope and the SEQUOIA 3 mm focal plane array, we have searched for molecular line emission from two atomic clouds associated with the Fermi Bubble of the Milky Way. Neither¹²CO nor¹³CO J=1–0 emission is detected from the HIcloud, MW-C20.¹²CO J=1–0 emission is detected from MW-C21 that is distributed within 11 clumps with most of the CO luminosity coming from a single clump. However, we find no¹³CO emission to a 3σbrightness temperature limit of 0.3 K. Using this limit andRADEXnon-local thermodynamic equilibrium (non-LTE) excitation models, we derive H₂column density upper limits of (0.4–3)×10²¹cm⁻²for a set of physical conditions and a H₂to¹²CO abundance ratio of 10⁴. Model CO-to-H₂conversion factors are derived for each set of physical conditions. We find the maximum value is 1.6×10²⁰cm⁻²/(K km s⁻¹). Increasing [H₂/¹²CO] to 10⁵to account for photodissociation and cosmic ray ionization increases the column density and X(CO) upper limits by a factor of 10. Applying these X(CO) limits to the CO luminosities, the upper limit on the total molecular mass in MW-C21 is 132±2 M_⊙, corresponding to <27% of the neutral gas mass. For the three clumps that are fully resolved, lower limits to the virial ratios are 288±32, 68±28, and 157±39, which suggest that these structures are bound by external pressure to remain dynamically stable over the entrainment time of 2×10⁶years or are being disrupted by shear and expansion over the clump crossing times of 3–8×10⁵years. The observations presented in this study add to the growing census of cold gas entrained within the Galactic Center wind.