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We have obtained a moderate-resolution spectrum of the quasar PKS 0528-250 with the Red Channel Spectrograph on the Multiple Mirror Telescope (MMT) in order to study the damped Ly$\alpha$ quasar absorption line system at z = 2.8115, which has 2 velocity components, at z = 2.8108 and z = 2.8132. We obtain a tentative detection of CO A-X $\lambda =1447.3$ \AA\ absorption from the z = 2.8132 velocity component. %The lack of H$_2$ absorption for this component H$_2$ absorption has been found at z = 2.8118 and thus appears not to be directly associated with the CO: this makes the appearance of CO difficult to understand. A measurement of C I $\lambda$ 1560 at z = 2.8115 can be used to constrain further the conditions in the H$_2$-containing gas. We obtain a new upper limit for the CO column density for the z = 2.8108 velocity component that does show H$_2$ absorption. Millimeter-wave observations obtained at the NRAO 12 meter telescope provide new upper limits on CO (J = 3-2) emission. However, these limits are not sensitive enough to rule out the possibility of the presence of a hyperluminous IR galaxy like IRAS F10214+4724.
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