A type of stable and identical two dimensional (2D) Au nanoclusters are discovered to exist on the Si(111)- root 3 x root 3 -Ag surface at submonolayer Au coverage. Self-assembly of these Au nanoclusters leads to a 2D superstructure Si(111)-root 21 x root 21 -(Ag+Au). This discovery helps us to obtain definite evidence that all the previously proposed atomic structure models for the root 21 x root 21 structure can be excluded. A model in which three Au adatoms locate on Ag triangles in the root 21 x root 21 unit cell is built based on the low-temperature scanning tunneling microscopy observations. Additionally, the stability and the uniformity of the Au nanocluster are discussed in terms of a substrate-mediated interaction among the Au adatoms, and the self-assembling process is understood by analogizing the Au nanocluster with a kind of surface molecule.