Temperature-dependent x-ray diffraction of the low-dimensional spin-1/2 quantum magnet TiOCl shows that the phase transition at T-c2=90 K corresponds to a lowering of the lattice symmetry. Below T-c1=66 K a twofold superstructure develops, that indicates the formation of spin-singlet pairs via direct exchange between neighboring Ti atoms, while the role of superexchange is found to be negligible. TiOCl thus is identified as a spin-Peierls system of pure one-dimensional chains of atoms. The first-order character of the transition at T-c1 is explained by the competition between the structurally deformed state below T-c2 and the spin-Peierls state below T-c1.