We investigate the spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. This problem arises in the study of the spectrum of the Dirichlet Laplacian in thick polyhedral domains having some symmetries such as the so-called Fichera layer. The geometry we consider depends on two parameters gathered in some vector κ = (κ_1,κ_2) which characterizes the domain at the edges. We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to κ. By exchanging the axes and/or modifying their orientations if necessary, it is sufficient to restrict the analysis to the cases κ_1\ge0 and κ_2∈ [−κ_1,κ_1]. We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to κ. In particular, we show that for a given κ_1 > 0, there is some h(κ_1) > 0 such that discrete spectrum exists for κ_2 ∈ (−κ_1,0) ∪ (h(κ_1),κ_1) whereas it is empty for κ_2 ∈ [0; h(κ_1)]. The proofs rely on classical arguments of spectral theory such as the max-min principle. The main originality lies rather in the delicate use of the features of the geometry.