<p>Ice deposits at the lunar poles are concentrated within permanently shadowed regions where water molecules are thermally stable. The geographic distribution of the deposits has been predicted and observed. However, the timing of their formation remains poorly constrained. Owing to the gradual decrease in the lunar obliquity with time, the cold-trapping regions have expanded monotonically since the passage of the Moon through the Cassini state transition ~4 Gyr ago. Here, using reflected ultraviolet starlight, we show with observations from the Lyman-Alpha Mapping Project that there is a strong correlation between the exposed-ice fraction and the age of the permanent shadow within which it resides. A model of water delivery, burial and loss predicts such a correlation if these processes operate on timescales that are comparable with the ages of the permanently shadowed regions. The exposed-ice area ratio of ~3.4% in the youngest permanently shadowed regions aged ~100 Myr favours models with high proportional loss rates. The results indicate that polar ice has accumulated quasi-continuously at least over the last ~1.5 Gyr rather than from a discrete event.</p>