Alternating current (ac) magnetic susceptibility and magnetization hysteresis loop measurements have been carried out for anionic bis(phthalocyaninato)terbium and bis(phthalocyaninato)dysprosium. The two mononuclear lanthanide complexes show the characteristic temperature and frequency dependence in the ac susceptibility signals, reflecting their slow magnetization relaxation. From the Arrhenius analysis of the ac susceptibility data obtained for a diluted sample in a diamagnetic matrix, it has been found that the magnetization relaxation in the Tb complex is dominated by the two-phonon Orbach process in the temperature range 2540 K and direct or Raman process below 25 K. In the Dy complex case, the Orbach process is the main relaxation process in the range 3-12 K. The Delta values in the Orbach term, corresponding to the height of the potential energy barrier to magnetic moment reversal, are in good agreement with the energy differences between the lowest and second lowest substates of the ground multiplet in the two cases. In the magnetization-field (M-H) loop measurements at 1.7 K, clear hysteresis has been observed for both complexes. These results indicate that the two double-decker phthalocyanine-lanthanide complexes behave as magnets at the single-molecular level. They are the first lanthanide compounds as well as the first mononuclear complexes showing such behaviors. Differences in the magnetization relaxation mechanism between the new "mononuclear lanthanide magnets" and the transition-metal-cluster-based SMMs (single-molecule magnets) are discussed.