4 pp.-- PACS: 73.20.-r, 73.21.-b, 79.60.Dp. The potassium monolayer on Cu(111) defines the simplest metallic quantum well that confines a single s-like discrete level. The analysis of the metallization onset in such a K adlayer reveals, however, a subtle electronic structure. The metallic monolayer condensate is actually characterized by a pair of two-dimensional states that lie below the Fermi energy, namely, a quantum well state and a resonant band reminiscent of the Cu(111) surface state. All quantum well states, resonances, and Cu substrate bulk bands exhibit smooth K coverage dependence, suggesting that changes in the crystal potential upon K adsorption extend from the surface and/or interface inside the Cu substrate. Peer reviewed