  A new precision measurement of the parity violating analyzing power in
longitudinally polarized electron scattering from the proton at very low Q^2 at
an incident energy of 1.16 GeV is in the final stages of preparation for
execution at Jefferson Laboratory (JLab). A 2200 hour measurement of the parity
violating asymmetry in elastic electron-proton scattering at Q^2 = 0.03
(GeV/c)^2 employing 180 microamp of 85% polarized beam on a 0.35 m long liquid
hydrogen target will determine the weak charge of the proton, Q_w = 1 -
4sin^2(theta_W), with 4% combined statistical and systematic errors. The
Standard Model makes a firm prediction of Q_w, based on the `running' of the
weak mixing angle sin^2(theta_W) from the Z-pole down to lower energies. Any
significant deviation of sin^2(theta_W) from its Standard Model prediction at
low Q^2 would constitute a signal of new physics. In the absence of new
physics, the envisaged experiment will provide a 0.3% determination of
sin^2(theta_W), making this a very competitive measurement of the weak mixing
angle. Complementary to the present experiment is a measurement of the weak
charge of the electron in parity violating Moller scattering at 11 GeV,
currently under consideration, with the upgraded CEBAF at JLab. The objective
of that experiment would be a measurement of sin2(theta_W) with a precision
comparable to or better than any individual measurement at the Z-pole.
