We studied the structural evolution of a 270-atom Ag-Au bimetallic nanoparticle (2 nm in size) with varying composition and temperature. The liquid to solid transition region and the solid-state structure were investigated using molecular dynamics simulations. To determine the exact transition temperature region, we applied the mean square displacement and structure deviation methods, as well as the generally used caloric curve of potential energy versus temperature. The results showed that a complete solid-solution phase diagram of the binary Ag-Au system was obtained. Irrespective of the composition, the freezing temperature of a Ag-Au bimetallic nanoparticle was lower than that of the bulk state by a margin of several hundred degrees, and three different solid-state structures are proposed in relation to the Au composition. Our phase diagram offers guidance for the application of Ag-Au nanoparticles.