  We report a new, high-precision measurement of the proton elastic form factor
ratio \mu_p G_E/G_M for the four-momentum transfer squared Q^2 = 0.3-0.7
(GeV/c)^2. The measurement was performed at Jefferson Lab (JLab) in Hall A
using recoil polarimetry. With a total uncertainty of approximately 1%, the new
data clearly show that the deviation of the ratio \mu_p G_E/G_M from unity
observed in previous polarization measurements at high Q^2 continues down to
the lowest Q^2 value of this measurement. The updated global fit that includes
the new results yields an electric (magnetic) form factor roughly 2% smaller
(1% larger) than the previous global fit in this Q^2 range. We obtain new
extractions of the proton electric and magnetic radii, which are
<r^2_E>^(1/2)=0.875+/-0.010 fm and <r^2_M>^(1/2)=0.867+/-0.020 fm. The charge
radius is consistent with other recent extractions based on the electron-proton
interaction, including the atomic hydrogen Lamb shift measurements, which
suggests a missing correction in the comparison of measurements of the proton
charge radius using electron probes and the recent extraction from the muonic
hydrogen Lamb shift.
