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Topological magnets are a new family of quantum materials providing great potential to realize emergent phenomena, such as the quantum anomalous Hall effect and the axion-insulator state. Here, we present our discovery that the stoichiometric ferromagnet MnBi_{8}Te_{13} with natural heterostructure MnBi_{2}Te_{4}/(Bi_{2}Te_{3})_{3} is an unprecedented “half-magnetic topological insulator,” with the magnetization existing at the MnBi_{2}Te_{4} surface but not at the opposite surface terminated by triple Bi_{2}Te_{3} layers. Our angle-resolved photoemission spectroscopy measurements unveil a massive Dirac gap at the MnBi_{2}Te_{4} surface and a gapless Dirac cone on the other side. Remarkably, the Dirac gap (about 28 meV) at the MnBi_{2}Te_{4} surface decreases monotonically with increasing temperature and closes right at the Curie temperature, thereby representing the first smoking-gun spectroscopic evidence of a magnetization-induced topological surface gap among all known magnetic topological materials. We further demonstrate theoretically that the half-magnetic topological insulator is desirable to realize the surface anomalous Hall effect, which serves as direct proof of the general concept of axion electrodynamics in condensed matter systems.