In this paper a criterion for the applicability of the finite section method to Wiener-Hopf integral operators W(a) with piecewise continuous symbol a acting on \(L^ p\) over the quarter-plane is established. It is shown that the finite section method is applicable to W(a) if and only if W(a) and its three associated operators \(W(a_ 1)\), \(W(a_ 2)\), \(W(a_{12})\) are invertible on \(L^ p({\mathbb{R}}^ 2_{++})=L^ p({\mathbb{R}}_+)\otimes L^ p({\mathbb{R}}_+)\) and if W(a) is Fredholm on all the spaces \(L^ r({\mathbb{R}}_+)\otimes L^ s({\mathbb{R}}_+)\) with mixed norm, where r,s\(\in [p,p/(p-1)]\).