We report on molecular self-assembly of biphenyl-4,4'-dicarboxylic acid (BPDCA) on CaCO3(10 (1) over bar4) under ultra-high vacuum conditions. Two-dimensional, ordered islands are obtained upon deposition at room temperature, coexisting with a streaky structure that is ascribed to individual, mobile molecules forming a two-dimensional gas-like phase. High-resolution non-contact atomic force microscopy (NC-AFM) images of the molecular islands reveal an ordered inner structure that is dominated by rows of molecules aligned side by side running along the [(42) over bar 61] crystallographic direction. A detailed analysis of these rows exhibits inter-row distances that are multiples of the calcite unit cell dimension along the [01 (1) over bar0] direction, clearly demonstrating the templating effect of the substrate. Our results indicate that an excellent size match of the molecular structure with respect to the underlying substrate results in an increased binding of the BPDCA molecules to the surface. In between the rows, a different molecular structure is coexisting with the molecules aligning head to tail. This structure is explained by intermolecular hydrogen bond formation very similar to the BPDCA bulk structure. The coexistence of the bulk-like structure with the row structure suggests a close balance of intermolecular and molecule-surface interactions to be responsible for the observed structure formation.