We use spin-polarized scanning tunneling microscopy and density functional theory (DFT) to study domain walls (DWs) and the Dzyaloshinskii-Moriya interaction (DMI) in epitaxial films of Co/Ir(111) and Pt/Co/Ir(111). Our measurements reveal DWs with fixed rotational sense for one monolayer of Co on Ir, with a wall width around 2.7 nm. With Pt islands on top, we observe that the DWs occur mostly in the uncovered Co/Ir areas, suggesting that the wall energy density is higher in Pt/Co/Ir(111). From DFT we find an interfacial DMI that stabilizes Néel-type DWs with clockwise rotational sense. The calculated DW widths are in good agreement with the experimental observations. The calculated total DMI nearly doubles from Co/Ir(111) to Pt/Co/Ir(111); however, in the latter case the DMI is almost entirely due to the Pt with only a minor Ir contribution. Therefore a simple additive effect, in which both interfaces contribute significantly to the total DMI, is not observed for one atomic Co layer sandwiched between Ir and Pt.