This paper is concerned with the study of invertibility properties of a singular integral operator naturally associated with the Zaremba problem for the Laplacian in infinite sectors in two dimensions, when considering its action on an appropriate Lebesgue scale of p integrable functions, for 1 p spectrum of this operator for each p ∈ (1, ∞). This analysis, along with a divergence theorem with non-tangential trace, are then used to establish well-posedness of the Zaremba problem with Lp data in infinite sectors in ℝ2.