We report a detailed theoretical study of the bonding and conduction properties of an hydrogen molecule joining either platinum or palladium electrodes. We show that an atomic arrangement where the molecule is placed perpendicular to the electrodes is unstable for all distances between electrodes. In contrast, the configuration where the molecule bridges the electrodes is stable in a wide range of distances. In this last case the bonding state of the molecule does not hybridize with the leads and remains localized within the junction. As a result, this state does not transmit charge so that electronic transport is carried only through the antibonding state. This fact leads to conductances of 1G(0) at most, where G(0)=2e(2)/h. We indeed find that G is equal to 0.9 and 0.6G(0) for Pt and Pd contacts, respectively.