  The ratio of the electric and magnetic form factor of the proton, $\mu_p
G_E^p/G_M^p$, has been measured for elastic electron-proton scattering with
polarized beam and target up to four-momentum transfer squared, $Q^2=5.66$
(GeV/c)$^2$ using the double spin asymmetry for target spin orientation aligned
nearly perpendicular to the beam momentum direction.
  This measurement of $\mu_p G_E^p/G_M^p$ agrees with the $Q^2$ dependence of
previous recoil polarization data and reconfirms the discrepancy at high $Q^2$
between the Rosenbluth and the polarization-transfer method with a different
measurement technique and systematic uncertainties uncorrelated to those of the
recoil-polarization measurements. The form factor ratio at $Q^2$=2.06
(GeV/c)$^2$ has been measured as $\mu_p G_E^p/G_M^p = 0.720 \pm 0.176_{stat}
\pm 0.039_{sys}$, which is in agreement with an earlier measurement with the
polarized target technique at similar kinematics. The form factor ratio at
$Q^2$=5.66 (GeV/c)$^2$ has been determined as $\mu_p
G_E^p/G_M^p=0.244\pm0.353_{stat}\pm0.013_{sys}$, which represents the highest
$Q^2$ reach with the double spin asymmetry with polarized target to date.
