Recently demonstrated superconducting atom-chips provide a platform for trapping atoms and coupling them to solid-state quantum systems. Controlling these devices requires a full understanding of the supercurrent distribution in the trapping structures. For type-II superconductors, this distribution is hysteretic in the critical state due to the partial penetration of the magnetic field in the thin superconducting film through pinned vortices. We report here an experimental observation of this memory effect. Our results are in good agreement with the redictions of the Bean model of the critical state without adjustable parameters. The memory effect allows to write and store permanent currents in micron-sized superconducting structures and paves the way towards new types of engineered trapping potentials.