  Spin waves in perpendicularly-magnetized Pt/Co/AlO$_x$/Pt ultrathin films
with varying Co thicknesses (0.6-1.2 nm) have been studied with Brillouin light
spectroscopy in the Damon-Eshbach geometry. The measurements reveal a
pronounced nonreciprocal propagation, which increases with decreasing Co
thicknesses. This nonreciprocity is attributed to an interfacial
Dzyaloshinskii-Moriya interaction (DMI), which is significantly stronger than
asymmetries resulting from surface anisotropies for such modes. Results are
consistent with an interfacial DMI constant $D_s = -1.7 \pm 0.11$ pJ/m, which
favors left-handed chiral spin structures. This suggests that such films below
1 nm in thickness should support novel chiral states like skyrmions.
