Based on first-principles density-functional calculations, we propose a novel growth mechanism of the 1D molecular lines on the H-terminated Si(001) surface where the line is directed across the Si dimer rows. The proposed structural model of the allyl mercaptan (ALM) line shows that the molecules adsorb across two Si dimers in the adjacent dimer rows with the Si-C and Si-S bonds, thereby yielding a higher thermodynamic stability compared to other alkene lines (containing a single Si-C bond per molecule) grown along the dimer rows. This accounts for a successful growth of ALM lines which were observed to be stable even at a high temperature of 650 K.