It is a "Great Seal" US Army button, in service from 1902 to the present day. That being said, the 1902-1910 version did not have a raised rim, and the World War One to 1923 version had "black finish" on its front to keep its shiny brass from giving away a hidden soldier's position. Your Great Seal button has a raised rim, and doesn't have "black finish," so it dates from sometime after 1923.
Some can be more narrowly time-dated by the maker's-mark on the back. Does yours have a name on its back?