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Correlations between magnetism and morphology of iron nanostructures of monatomic height on Pt(997) substrates are studied using x-ray magnetic circular dichroism as well as scanning tunneling microscopy and helium scattering. A drastic collapse of the average magnetization by more than a factor of 4 is observed when increasing the iron coverage from 0.1 to 0.2 ML. This effect goes along with a softening of the magnetic anisotropy energy and a gradual reorientation of the magnetic easy axis from in plane to out of plane. The experimental findings together with electronic density-functional calculations suggest the formation of a complex magnetic phase in corrugated rim regions of Fe islands, leading to both ferromagnetic and antiferromagnetic exchange couplings of Fe moments depending on their various local bonding configurations.