  The electromagnetic form factors of the nucleon characterize the effect of
its internal structure on its response to an electromagnetic probe as studied
in elastic electron-nucleon scattering. These form factors are functions of the
squared four-momentum transfer $Q^2$ between the electron and the proton. The
two main classes of observables of this reaction are the scattering cross
section and polarization asymmetries, both of which are sensitive to the form
factors in different ways. When considering large momentum transfers,
double-polarization observables offer superior sensitivity to the electric form
factor. This thesis reports the results of a new measurement of the ratio of
the electric and magnetic form factors of the proton at high momentum transfer
using the recoil polarization technique. A polarized electron beam was
scattered from a liquid hydrogen target, transferring polarization to the
recoiling protons. These protons were detected in a magnetic spectrometer which
was used to reconstruct their kinematics, including their scattering angles and
momenta, and the position of the interaction vertex. A proton polarimeter
measured the polarization of the recoiling protons by measuring the azimuthal
asymmetry in the angular distribution of protons scattered in CH$_2$ analyzers.
The scattered electron was detected in a large-acceptance electromagnetic
calorimeter in order to suppress inelastic backgrounds. The measured ratio of
the transverse and longitudinal polarization components of the scattered proton
is directly proportional to the ratio of form factors $G_E^p/G_M^p$. The
measurements reported in this thesis took place at $Q^2=$5.2, 6.7, and 8.5
GeV$^2$, and represent the most accurate measurements of $G_E^p$ in this $Q^2$
region to date.
