Single crystals of the Pb2Fe2−xMnxGe2O9 (x = 0.16) antiferromagnet have been grown. Using the specific heat measurements, a Néel temperature of TN = (42.0 ± 0.5) K for the synthesized crystals has been found. It has been shown using the magnetic measurements that, due to the competition between the magnetoanisotropic contributions of the iron and manganese subsystems in the crystals, near a temperature of Tc = 22 K, a spontaneous spin-reorientation transition occurs, the temperature of which in an applied magnetic field changes with the field value and orientation relative to the rhombic axes of the crystal. Based on the analysis of the temperature and field dependences of the magnetization obtained at different orientations of the magnetic field, it has been established that, below Tc, an inclined magnetic structure is formed in the crystal. The antiferromagnetic vector of the inclined structure rotates smoothly in the rhombic bc plane with increasing temperature from a direction close to the b axis at T = 4.2 K and tends to the rhombic c axis at T = Tc. The rotation of the antiferromagnetic vector occurs also at fixed temperatures T < Tc with increasing magnetic field. In the temperature range of Tc < T < TN, the antiferromagnetic vector is oriented along the rhombic c axis. Magnetic phase diagrams of states have been built for different magnetic field orientations relative to the rhombic axes of the crystal. The richest phase diagram is shown to correspond to the orientation H||c and contains, along with the above-listed states, one more inclined phase, in which the antiferromagnetic vector rotates toward the rhombic a axis direction with a change in temperature or magnetic field.