The axion is a hypothetical particle proposed to solve the strong CP problem in quantum chromodynamics of particle physics and possesses a cosmological implication as cold dark matter if its mass falls in a certain range. A conventional axion dark matter search experiment employs a microwave resonant cavity in a strong magnetic field to enhance the feeble signal. To explore high mass regions, a multiple-cavity design has been considered to increase the detection volume for a given magnet bore. We propose a new cavity design, referred to as multiple-cell cavity, to provide a highly efficient approach to high mass regions compared to the multiple-cavity design, by means of larger detection volume, simpler detector setup, and unique phase-matching mechanism. We discuss the characteristics of this concept and demonstrate the experimental feasibility. We also present the first results of a search for invisible axion dark matter using a double-cell cavity and set new limits on the axion-photon coupling over the mass range between 13.0 and 13.9$\,\mu$eV. The results not only demonstrates the novelty of the cavity concept for high-mass axion searches, but also suggests it can make considerable contributions to the next-generation experiments.