Quantum memory is one of the basic building blocks in large-scale quantum computers. In the paper, we propose a scheme to realize a quantum memory in a hybrid system by controllable Stark-chirped rapid adiabatic passages. In the hybrid system, by taking advantage of long coherence times in microscopic system, which is a twolevel system naturally embedded in a current-biased Josephson junction, individual two-level system can be regarded as a quantum memory. The scheme of storage and retrieval is realized by Stark-chirped rapid adiabatic passages, which is insensitive to the details of the applied adiabatic pulses. In numerical investigation of the storage process using adiabatic master equations, we demonstrate that high-fidelity quantum memory can be achieved under practical noises. Finally, the experimental feasibility and performance are discussed based on the current experimental status.