We have studied the influence of preadsorbed CO on the dissociative adsorption of H-2 on Ru(0001) with density functional theory calculations. For a coverage of 1/3 ML CO, we investigated different possible reaction paths for hydrogen dissociation using nudged elastic band and adaptive nudged elastic band calculations. One reaction path was studied in detail through an energy decomposition and molecular orbital type of analysis. The minimum barrier for H-2 dissociation is found to be 0.29 eV. At the barrier the H-H bond is hardly stretched. Behind this barrier a molecular chemisorption minimum is present. Next, the molecule overcomes a second barrier, with a second local chemisorption minimum behind it. To finally dissociate to chemisorbed atoms, the molecule has to overcome a third barrier. To move along the reaction path from reactants to products, the hydrogen molecule needs to rotate, and to significantly change its center-of-mass position. The procedure of mapping out reaction paths for H-2 reacting on low-index surfaces of bare metals (computing two-dimensional elbow plots for fixed impact high-symmetry sites and H-2 orientations parallel to the surface) does not work for H-2+CO/Ru. The first barrier in the path is recovered, but the features of the subsequent stretch to the dissociative chemisorption minimum are not captured, because the molecule is not allowed to change its center-of-mass position or to rotate. The dissociative chemisorption of H-2 on CO/Ru(0001) is endoergic, in contrast to the case of H-2 on bare Ru(0001). The zero-point energy corrected energies of molecularly and dissociatively chemisorbed H-2 are very close, suggesting that it may be possible to detect molecularly chemisorbed H-2 on (root 3>x root 3)R30 degrees CO/Ru(0001). The presence of CO on the surface increases the barrier height to dissociation compared with bare Ru(0001). Based on an energy decomposition and molecular orbital analysis we attribute the increase in the barrier height mainly to an occupied-occupied interaction between the bonding H-2 sigma(g) orbital and the (surface-hybridized) CO 1 pi orbitals, i.e., to site blocking. There is a small repulsive contribution to the barrier from the interaction between the H-2 molecule and the Ru part of the CO covered Ru surface, but it is smaller than one might expect based on the calculations of H-2 interacting with a clean Ru surface, and on calculations of H-2 interacting with the CO overlayer only. Actually, the analysis suggests that the Ru surface as a subsystem is (slightly) more reactive for the reaction path studied with CO preadsorbed on it than without it. Thus, the results indicate that the influence of CO on H-2 dissociation on Ru is not only a simple site-blocking effect, the electronic structure of the underlying Ru is changed