  We match the known chiral perturbation theory representation of the pi pi
scattering amplitude to two loops with a phenomenological description that
relies on the Roy equations. On this basis, the corrections to Weinberg's low
energy theorems for the S-wave scattering lengths are worked out to second
order in the expansion in powers of the quark masses. The resulting
predictions, a00=0.220 \pm 0.005, a20=-0.0444 \pm 0.0010, contain remarkably
small uncertainties and thus allow a very sensitive experimental test of the
hypothesis that the quark condensate is the leading order parameter of the
spontaneously broken chiral symmetry.
