We present the first results of a search for invisible axion dark matter using a multiple-cell cavity haloscope. This cavity concept was proposed to provide a highly efficient approach to high-mass regions compared to the conventional multiple-cavity design, with larger detection volume, simpler detector setup, and a unique phase-matching mechanism. Searches with a double-cell cavity superseded previous reports for the axion-photon coupling over the mass range between 13.0 and 13.9 mu eV. This result 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.