A thermally programmable memory has a programmable element (20) of a thermally programmable resistance preferably of phase change material, material and a blown antifuse (80) located adjacent to the programmable material. Such a blown antifuse has a dielectric layer (100) surrounded by conductive layers (90, 110) to enable a brief high voltage to be applied across the dielectric to blow a small hole in the dielectric during manufacture to form a small conductive path which can be used as a tiny electrical heater for programming the material. Due to the current confinement by the hole, the volume of the material that must be heated in order to switch to a highly-resistive state is very small. As a result the programming power can be low.