Structural and thermodynamic properties of single-crystalline UNi1-xGe2 with x = 0.66 have been investigated by measuring magnetization, specific heat, and thermal expansion over a wide range of temperatures and magnetic fields. The measurements revealed the emergence of a long-range antiferromagnetic ordering of uranium magnetic moments below the Neel temperature T-N = 45.5(1) K and the existence of two easy axes in the studied compound, namely, b and c, which correspond to the plane of the uranium zigzag chains. Magnetic field applied along these two crystallographic directions induces in the system a first-order metamagnetic phase transition (from antiferromagnetism to field-polarized paramagnetism), and the width of the magnetic hysteresis associated with that transition reaches as much as about 40 kOe at the lowest temperatures. A magnetic phase diagram developed from the experimental data showed that the metastable region associated with that magnetic hysteresis forms a funnel that narrows toward the Neel point in a zero magnetic field. The four-layer Ising model has successfully predicted the collinear antiferromagnetic structure in UNi0.34Ge2 (known from earlier reports), its magnetic phase diagram, and temperature and field variations of its magnetization. Moreover, it suggests that the first-order phase transition extends down to a zero magnetic field, although it is barely detectable in the experiments performed in low magnetic fields. According to this model, the second-order phase transition occurs in the compound only in a zero field.