  Exclusive $\rho^0$-meson electroproduction is studied by the HERMES
experiment, using the 27.6 GeV longitudinally polarized electron/positron beam
of HERA and a transversely polarized hydrogen target, in the kinematic region
1.0 GeV$^2$<Q$^2$<7.0 GeV$^2$, 3.0 GeV<W<6.3 GeV, and -t'<0.4 GeV$^2$. Using an
unbinned maximum-likelihood method, 25 parameters are extracted. These
determine the real and imaginary parts of the ratios of several helicity
amplitudes describing $\rho^0$-meson production by a virtual photon. The
denominator of those ratios is the dominant amplitude, the
nucleon-helicity-non-flip amplitude $F_{0\frac{1}{2}0\frac{1}{2}}$, which
describes the production of a longitudinal $\rho^{0}$-meson by a longitudinal
virtual photon. The ratios of nucleon-helicity-non-flip amplitudes are found to
be in good agreement with those from the previous HERMES analysis. The
transverse target polarization allows for the first time the extraction of
ratios of a number of nucleon-helicity-flip amplitudes to
$F_{0\frac{1}{2}0\frac{1}{2}}$. Results obtained in a handbag approach based on
generalized parton distributions taking into account the contribution from pion
exchange are found to be in good agreement with these ratios. Within the model,
the data favor a positive sign for the $\pi-\rho$ transition form factor. By
also exploiting the longitudinal beam polarization, a total of 71 $\rho^0$
spin-density matrix elements is determined from the extracted 25 parameters, in
contrast to only 53 elements as directly determined in earlier analyses.
