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(lll) 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 (root 3 x root 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 degrees to each other.