Two structurally constrained chelators based on a fused bicyclic scaffold, 4‐amino‐4‐methylperhydro‐pyrido[1,2‐a][1,4]diazepin‐N,N′,N′‐triacetic acids [(4R*,10aS*)‐PIDAZTA (L1) and (4R*,10aR*)‐PIDAZTA (L2)], were designed for the preparation of GaIII‐based radiopharmaceuticals. The stereochemistry of the ligand scaffold has a deep impact on the properties of the complexes, with unexpected [Ga(L2)OH] species being superior in terms of both thermodynamic stability and inertness. This peculiar behavior was rationalized on the basis of molecular modeling and appears to be related to a better fit in size of GaIII into the cavity of L2. Fast and efficient formation of the GaIII chelates at room temperature was observed at pH values between 7 and 8, which enables 68Ga radiolabeling under truly physiological conditions (pH 7.4).