The first-principles FLAPW (full potential linearized augmented plane wave) electronic structure calculations were performed for the Ag(5)Li(8) gamma-brass, which contains 52 atoms in a unit cell and has been known for many years as one of the most structurally complex alloy phases. The calculations were also made for its neighboring phase AgLi B2 compound. The main objective in the present work is to examine if the Ag(5)Li(8) gamma-brass is stabilized at the particular electrons per atom ratio e/a = 21/13 in the same way as some other gamma-brasses like Cu(5)Zn(8) and Cu(9)Al(4), obeying the Hume-Rothery electron concentration rule. For this purpose, the e/a value for the Ag(5)Li(8) gamma-brass as well as the AgLi B2 compound was first determined by means of the FLAPW-Fourier method we have developed. It proved that both the gamma-brass and the B2 compound possess an e/a value equal to unity instead of 21/13. Moreover, we could demonstrate why the Hume-Rothery stabilization mechanism fails for the Ag(5)Li(8) gamma-brass and proposed a new stability mechanism, in which the unique gamma-brass structure can effectively lower the band-structure energy by forming heavily populated bonding states near the bottom of the Ag-4d band.