We investigate the spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. The geometry 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. 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.