Infrared interferometry of supergiant and Mira stars has recently been reinterpreted as revealing the presence of deep molecular layers. Empirical models for a photosphere surrounded by a simple molecular layer or envelope have led to a consistent interpretation of previously inconsistent data. The stellar photospheres are found to be smaller than previously understood, and the molecular layer is much higher and denser than predicted by hydrostatic equilibrium. However, the analysis was based on spatial observations with medium-band optical filters, which mixed the visibilities of different spatial structures. This paper reports spatial interferometry with narrow spectral bands, isolating near-continuum and strong molecular features, obtained for the supergiant mu Cep. The measurements confirm strong variation of apparent diameter across the K-band. A layer model shows that a stellar photosphere of angular diameter 14.11 +/- 0.60 mas is surrounded by a molecular layer of diameter 18.56 +/- 0.26 mas, with an optical thickness varying from nearly zero at 2.15 mu m to > 1 at 2.39 mu m. Although mu Cep and alpha Ori have a similar spectral type, interferometry shows that they differ in their radiative properties. Comparison with previous broad-band measurements shows the importance of narrow spectral bands. The molecular layer or envelope appears to be a common feature of cool supergiants.