We propose a new precision measurement of parity violating electron scattering on the proton at very low Q² and forward angles to challenge predictions of the Standard Model and search for new physics. A unique opportunity exists to carry out the first precision measurement of the proton's weak charge, Q^P_W=1 - 4sin²_W, at JLab, building on technical advances that have been made in the laboratory's world-leading parity violation program and using the results of earlier experiments to constrain hadronic corrections. A 2200 hour measurement of the parity violating asymmetry in elastic ep scattering at Q²=0.03 (GeV/c)² employing 180 A of 80% polarized beam on a 35 cm liquid Hydrogen target will determine the proton's weak charge with 4% combined statistical and systematic errors. The Standard Model makes a firm prediction of Q^p_W, based on the running of the weak mixing angle sin²_W from the Z⁰ pole down to low energies, corresponding to a 10 effect in our experiment. Any significant deviation of sin²_W from the Standard Model prediction at low Q² would be a signal of new physics, whereas agreement would place new and significant constraints on possible Standard Model extensions. In the absense of physics beyond the Standard Model, our experiment will provide a 0.3% measurement of sin²_W, making this a very competitive standalone measurement of the weak mixing angle.