The bistability of ordered spin states in ferromagnets provides the basis for magnetic memory functionality. The latest generation of magnetic random access memories rely on an efficient approach in which magnetic fields are replaced by electrical means for writing and reading the information in ferromagnets. This concept may eventually reduce the sensitivity of ferromagnets to magnetic field perturbations to being a weakness for data retention and the ferromagnetic stray fields to an obstacle for high-density memory integration. Here we report a room-temperature bistable antiferromagnetic (AFM) memory that produces negligible stray fields and is insensitive to strong magnetic fields. We use a resistor made of a FeRh AFM, which orders ferromagnetically roughly 100 K above room temperature, and therefore allows us to set different collective directions for the Fe moments by applied magnetic field. On cooling to room temperature, AFM order sets in with the direction of the AFM moments predetermined by the field and moment direction in the high-temperature ferromagnetic state. For electrical reading, we use an AFM analogue of the anisotropic magnetoresistance. Our microscopic theory modelling confirms that this archetypical spintronic effect, discovered more than 150 years ago in ferromagnets, is also present in AFMs. Our work demonstrates the feasibility of fabricating room-temperature spintronic memories with AFMs, which in turn expands the base of available magnetic materials for devices with properties that cannot be achieved with ferromagnets.