  We demonstrate theoretically the existence of a linear dc Stark shift of the
individual substates of an alkali atom in its ground state, dressed by a
circularly polarized laser field. It arises from the electroweak nuclear
anapole moment violating P but not T. It is characterized by the pseudoscalar
equal to the mixed product formed with the photon angular momentum and static
electric and magnetic fields. We derive the relevant left-right asymmetry with
its complete signature in a field configuration selected for a precision
measurement with cold atom beams. The 3,3 to 4,3 Cs hyperfine-transition
frequency shift amounts to 7 $\mu$Hz for a laser power of about 1 kW at 877 nm,
E=100 kV/cm and B larger than 0.5 G.
