Four homogeneous solid solutions were found in the silver-magnesium alloy system. The α-primary solid solution of magnesium in silver has a f.c.c. structure. The β-solid solution has a b.c.c. structure. The γ-solid solution appears to have a complicated structure. While others have identified it as hexagonal, no definite structure could be assigned to it from our diffraction data. The δ-solid solution has a h.c.p. structure of magnesium. The lattic parameter a0 in the α-solid solution increases with increasing magnesium concentration but exhibits a marked negative deviation from Vegard's Law. The a0 in the β-solid solution, however, increases linearly with increasing magnesium concentration. Silver and magnesium atoms are distributed at random in the α-phase but take up preferred positions in the β-phase. The ordering process takes place throughout the β-phase, but is most nearly complete in the region of 50 atomic percent Mg.