The history of exposure to cosmic rays was studied for nine basalt rocks based on the cosmic-ray-produced noble gas isotopes. The results for the five basalts 10003, 10047, 12004, 12051, and 77135 are consistent with a single-state exposure. Four basalts, 10057, 10069, 12040, and 12053, experienced a multistage exposure. Using the 235U-136Xe g dating method, the time of their early exposure to cosmic rays on the lunar surface was dated to have begun 3--4 aeons ago, immediately after rock formation as defined by their crystallization ages. The durations and shielding depths for the proposed two-stage exposure models of the four basalts were calculated. From the shielding at the early stage we estimate a lower limit of 50 cm for the thickness of the lava flows from which the investigated basalts originate.