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An X-ray diffraction analysis of a two-dimensional organic structure is presented. Direct evidence for the singly bonded 5-5'-diuracil-dimer is given which is the object of intense discussion in biochemistry. It is formed in ultra-high-vacuum by deposition of 5-I-uracil on a clean Ag(111) surface kept at 390 K. The dimers are arranged in an uniaxial incommensurate superstructure with two molecules per unit cell whereas the released I-atoms form a (√3x√3) superstructure. The molecular structure is characterized by parallel chains of hydrogen bonded dimers. The individual pyrimidine rings of the dimers are inclined by about 30-40 o to each other.