A system and method for writing to a magnetic memory written in a thermally assisted manner, each memory point formed by a magnetic tunnel junction, and having a substantially circular cross-section of the memory which is parallel to the plane of the layers forming the tunnel junction. The tunnel junction includes at least a trapped layer with a fixed magnetisation direction, a free layer with a variable magnetisation direction with an insulating layer arranged there between. The free layer is formed from at least one soft magnetic layer and a trapped layer, with the two layers being magnetically coupled by contact. During read operations and at rest, the operating temperature of the memory is lower than the blocking temperature of the free and trapped layers, respectively.