We present a first-principles study of model domain walls (DWs) in prototypic ferroelectric PbTiO3. At high temperature the DW structure is somewhat trivial, with atoms occupying high-symmetry positions. However, upon cooling the DW undergoes a symmetry-breaking transition characterized by a giant dielectric anomaly and the onset of a large and switchable polarization. Our results thus corroborate previous arguments for the occurrence of ferroic orders at structural DWs, providing a detailed atomistic picture of a temperature-driven DW-confined transformation. Beyond its relevance to the field of ferroelectrics, our results highlight the interest of these DWs in the broader areas of low-dimensional physics and phase transitions in strongly fluctuating systems.