For the handling of high level nuclear waste (HLW), new glass formulations with a high waste capacity and an enhanced thermal stability, chemical durability, and processability are under consideration. This study focuses on the durability of peraluminous glasses in the SiO2–Al2O3–B2O3–Na2O–CaO–La2O3 system, defined by an excess of Al3+ ions compared with the network-modifying cations Na+ and Ca2+. To qualify the behavior of such a peraluminous glass in a geological storage situation, its chemical durability was studied in various environments (pure water, groundwater, and alkaline solutions related to a cement environment) and glass alteration regimes (initial rate, residual rate, and resumption of alteration). The alteration solution was characterized by inductively coupled plasma, and the altered glass by scanning electron microscopy, X-ray diffraction and secondary ion mass spectrometry. A comparative study of the chemical durability of these and reference glasses (ISG and SON68) over all timescales highlights the remarkable properties of the former. While their initial dissolution rate is of the same order as the reference glasses, the gel formed under silica saturation conditions is more passivating, making its dissolution rate at least one order of magnitude lower, while its low alkalinity makes it less susceptible to clayey groundwater and highly alkaline solutions.