We carry out a first-principles density functional study of the interaction between a monatomic Pt wire and a CO molecule, comparing the energy of different adsorption configurations (bridge, on top, substitutional, and tilted bridge) and discussing the effects of spin-orbit (SO) coupling on the electronic structure and on the ballistic conductance of two of these systems (bridge and substitutional). We find that, when the wire is unstrained, the bridge configuration is energetically favored, while the substitutional geometry becomes possible only after the breaking of the Pt-Pt bond next to CO. The interaction can be described by a donation/back-donation process similar to that occurring when CO adsorbs on transition-metal surfaces, a picture which remains valid also in presence of SO coupling. The ballistic conductance of the (tipless) nanowire is not much reduced by the adsorption of the molecule on the bridge and on-top sites, but shows a significant drop in the substitutional case. The differences in the electronic structure due to the SO coupling influence the transmission only at energies far away from the Fermi level so that fully- and scalar-relativistic conductances do not differ significantly.