Let W be a finite Coxeter group generated by real reflections in a complex vector space. We compute the integral cohomology of the Milnor fibre of the discriminant bundle Delta: C-n/W --> C, together with the action of the monodromy, for the whole list of exceptional groups. Here A is the map induced by the square of the polynomial defining the arrangement of reflection hyperplanes of W. The computation is equivalent to that of the cohomology, with suitable local coefficients, of the corresponding Artin group. These computations complete, for the exceptional cases, those performed by De Concini et al. for rational coefficients.