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We use UV measurements of interstellar CO towards nearby stars to calculatethe density in the diffuse molecular clouds containing the moleculesresponsible for the observed absorption. Chemical models and recentcalculations of the excitation rate coefficients indicate that the regions inwhich CO is found have hydrogen predominantly in molecular form. We carry outstatistical equilibrium calculations using CO-H2 collision rates to solve forthe H2 density in the observed sources without including effects of radiativetrapping. We have assumed kinetic temperatures of 50 K and 100 K, finding thischoice to make relatively little difference to the lowest transition. For thesources having T_ex(1-0) only, for which we could determine upper and lowerdensity limits, we find <n(H2)> = 49 cm-3. While we can find a consistentdensity range for a good fraction of the sources having either two or threevalues of the excitation temperature, there is a suggestion that the higher-Jtransitions are sampling clouds or regions within diffuse molecular cloudmaterial that have higher densities than the material sampled by the J = 1-0transition. The assumed kinetic temperature and derived H2 density areanticorrelated when the J = 2-1 transition data, the J = 3-2 transition data,or both are included. For sources with either two or three values of theexcitation temperature, we find average values of the midpoint of the densityrange that is consistent with all of the observations equal to 68 cm-3 for T_k= 100 K and 92 cm-3 for T_k = 50 K. The data for this set of sources imply thatdiffuse molecular clouds are characterized by an average thermal pressurebetween 4600 and 6800 Kcm-3.

Diffuse Molecular Cloud Densities from UV Measurements of CO Absorption