Restructuring processes of the Si(111) surface from the 7X7 to the square-root 3 X square-root 3 structure upon the deposition of Ag has been observed by a high-temperature in situ STM method. Although the square-root 3 X square-root 3 structure is formed at substrate temperatures above 240-degrees-C, Ag islands do not have the square-root 3 X square-root 3 structure at the initial stage of the condensation. The restructuring is found to occur after the Ag island has grown larger than the critical size, which is four times as large as the half unit cell of the 7X7 structure. The restructuring starts from the edges of such Ag islands and the area of the square-root 3 X square-root 3 structure expands gradually as they grow. The hole-island pairs are formed so as to conserve the number of Si atoms underneath each Ag island.