  We report the high-field induced magnetic phases and phase diagram of a high
quality \urxrs~single crystal prepared using a modified Czochralski method. Our
study, that combines high-field magnetization and electrical resistivity
measurements, shows for fields applied along the $c$-axis direction three
field-induced magnetic phase transitions at $\mu_{0} H_{c1}$ = 21.60 T,
$\mu_{0} H_{c2}$ = 37.90 T and $\mu_{0} H_{c3}$ = 38.25 T, respectively. In
agreement with a microscopic up-up-down arrangement of the U magnetic moments
the phase above $H_{c1}$ has a magnetization of about one third of the
saturated value. In contrast the phase between $H_{c2}$ and $H_{c3}$ has a
magnetization that is a factor of two lower than above the $H_{c3}$, where a
polarized Fermi-liquid state with a saturated moment $M_{s}$ $\approx$ 2.1
$\mu_{B}$/U is realized. Most of the respective transitions are reflected in
the electrical resistivity as sudden drastic changes. Most notably, the phase
between $H_{c1}$ and $H_{c2}$ exhibits substantially larger values. As the
temperature increases, transitions smear out and disappear above $\approx$ 15
K. However, a substantial magnetoresistance is observed even at temperatures as
high as 80 K. Due to a strong uniaxial magnetocrystalline anisotropy a very
small field effect is observed for fields apllied perpendicular to the $c$-axis
direction.
