We report on the formation of triple-atomic-layer metal films on Si(111) by deposition of Mg onto In double-layer films. The deposited Mg atoms are intercalated between the In layers and Si substrate and form a buffer layer with $(\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3})$ periodicity on an unreconstructed Si(111) surface. Two metal layers with nearly hexagonal close-packed arrangement are stacked in an ABC sequence on the buffer layer. The (In, Mg) triple-layer phase shows free-electron-like electronic structure with circular Fermi surface. Unlike the single Fermi circle of the In double layers, the Fermi surface of the (In, Mg) triple layers is composed of two circles with different radii. The larger and smaller Fermi circles are found to come from bonding and antibonding states between the top and middle layers of the three layers. The bottom layer acts as a buffer layer to saturate the Si dangling bonds and realize a nearly freestanding double-layer metal.