We report a polyol method for the facile synthesis of Ag nanocubes having sharp corners and edges, together with edge lengths below 15 nm. The rapid nucleation of Ag atoms was facilitated through the addition of a trace amount of SH- to generate Ag2S clusters while the corners and edges of the nanocubes were sharpened through the introduction of Br- as a regulator of the growth kinetics and a capping agent for the Ag(100) surface. Because of their much smaller size relative to the more commonly used capping agent based on poly(vinylpyrrolidone), Br- ions are more effective in passivating the {100} facets on very small Ag nanocubes. The mechanistic roles of these additives, along with the effects of their interactions with other species present in the reaction solution, were all systematically investigated. The concentration of SH- was found to be a particularly effective parameter for tuning the edge length of the nanocubes. As a result of the understanding gained during the course of this study, Ag nanocubes with uniform edge lengths controllable in the range of 13-23 nm could be reliably produced. The nanocubes of 13.4 +/- 0.4 nm in edge length constitute the smallest nanocrystals of this kind reported to date; they also possess sharper corners and edges relative to the limited examples of sub-20 nm Ag nanocubes reported in the literature. The availability of such small and sharp Ag nanocubes will open the door to an array of applications in plasmonics, catalysis, and biomedicine.