The in-plane orientation of the magnetization in the center of domain walls is measured in Co/Ir(111) as a function of Co thickness via scanning electron microscopy with polarization analysis. Uncapped, thermally evaporated cobalt on an Ir(111) single-crystal surface is imaged in situ in ultrahigh vacuum. The initial pseudomorphic growth with an atomically flat interface of cobalt on iridium ensures comparability to theoretical calculations and provides a study of an interface that is as ideal as possible. Below a cobalt thickness of 8.8 monolayers, the magnetic domain walls are purely Néel oriented and show a clockwise sense of rotation. For larger thicknesses the plane of rotation changes and the domain walls show a significant Bloch-like contribution, allowing one to calculate the strength of the Dzyaloshinskii-Moriya interaction (DMI) from energy minimization. From the angle between the plane of rotation and the domain-wall normal an interfacial DMI parameter Ds=−(1.07±0.05)pJ/m is determined, which corresponds to a DMI energy per bond between two Co atoms at the interface of dtot=−(1.04±0.05)meV.