  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 ultra-high 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 N\'eel 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 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 $D_s = -(1.07 \pm
0.05)$ pJ/m is determined, which corresponds to a DMI energy per bond between
two Co atoms at the interface of $d_{tot} = -(1.04 \pm 0.05)$ meV.
