This paper is dedicated to a description of the poles of the Igusa local zeta function Z(s, f, v) when f(x, y) satisfies a new non-degeneracy condition called arithmetic non-degeneracy. More precisely, we attach to each polynomial f(x, y) a collection of convex sets Γ A (f) = {Γ f,1 Γ f,l0 ) } called the arithmetic Newton polygon of f(x, y), and introduce the notion of arithmetic non-degeneracy with respect to Γ A (f). If L v is a p-adic field, and f(x, y) E L v [x, y] is arithmetically non-degenerate, then the poles of Z(s, f, v) can be described explicitly in terms of the equations of the straight segments that form the boundaries of the convex sets Γ f,i ....,Γ f,l0 . Moreover, the proof of the main result gives an effective procedure for computing Z(s, f, v).