We use UV measurements of interstellar CO towards nearby stars to calculate the density in the diffuse molecular clouds containing the molecules responsible for the observed absorption. Chemical models and recent calculations of the excitation rate coefficients indicate that the regions in which CO is found have hydrogen predominantly in molecular form. We carry out statistical equilibrium calculations using CO-H2 collision rates to solve for the H2 density in the observed sources without including effects of radiative trapping. We have assumed kinetic temperatures of 50 K and 100 K, finding this choice to make relatively little difference to the lowest transition. For the sources having T_ex(1-0) only, for which we could determine upper and lower density limits, we find <n(H2)> = 49 cm-3. While we can find a consistent density range for a good fraction of the sources having either two or three values of the excitation temperature, there is a suggestion that the higher-J transitions are sampling clouds or regions within diffuse molecular cloud material that have higher densities than the material sampled by the J = 1-0 transition. The assumed kinetic temperature and derived H2 density are anticorrelated 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 the excitation temperature, we find average values of the midpoint of the density range 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 that diffuse molecular clouds are characterized by an average thermal pressure between 4600 and 6800 Kcm-3.
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