Photoemission spectroscopy with synchrotron radiation has been used to study the valence-band behavior as well as the Au 4f core-level binding energy shifts of coadsorbed Au and Ag overlayers on Ru(001) at one-monolayer coverage each as monitored with thermal desorption measurements. The results were compared with bulk Au-Ag results obtained previously to determine if and under what conditions a two-dimensional alloy is formed. We found that alloying occurred to some extent at room temperature and a stable alloy was formed at elevated temperatures (800 K) when Ag was deposited first, while there was no alloying when Au was deposited first on the Ru substrate. It appears that there is a significant difference in the electronic structure of the overlayers depending on the order of the deposition. These results and their implications are discussed.