  It is demonstrated that Breit and negative-energy state contributions reduce
the 2.5 sigma deviation [S.C. Bennett and C.E. Wieman, Phys. Rev. Lett. 82,
2484 (1999)] in the value of the weak charge of 133Cs from the Standard Model
prediction to 1.7sigma. The corrections are obtained in the relativistic
many-body perturbation theory by combining all-order Coulomb and second-order
Breit contributions. The corrections to parity-nonconserving amplitudes amount
to 0.6% in 133Cs and 1.1% in 223Fr. The relevant magnetic-dipole hyperfine
structure constants are modified at the level of 0.3% in Cs, and 0.6% in Fr.
Electric-dipole matrix elements are affected at 0.1% level in Cs and a few 0.1%
in Fr.
