We present a combined surface x-ray diffraction (SXRD) and theoretical analysis of the geometric structure of nanometer sized Co islands deposited on Cu(001) at 170 K. Two-dimensional nanoislands consisting of only 20-40 atoms are characterized by a 4%-8% contraction of the interatomic distances as compared to the bulk (2.51 Å). This strongly exceeds "usual" lattice relaxations normally seen for surfaces. The SXRD analysis is based on the analysis of the registry of the Co adatoms relative to the (1x1) surface unit cell of the Cu(001) substrate crystal. Static displacements of the Co atoms of 0.18 Å (root mean square) out of the equilibrium hollow sites are observed which are in agreement with predictions based on molecular-dynamics calculations.