  We report a test of the universality of free fall (UFF) by comparing the
gravity acceleration of the $^{87}$Rb atoms in $m_F=+1$ versus that in
$m_F=-1$, where the corresponding spin orientations are opposite. A
Mach-Zehnder-type atom interferometer is exploited to sequentially measure the
free fall acceleration of the atoms in these two magnetic sublevels, and the
resultant E$\rm{\ddot{o}}$tv$\rm{\ddot{o}}$s ratio is ${\eta _S}
=(0.2\pm1.2)\times 10^{-7}$. This also gives an upper limit of $1.1\times
10^{-21}$ GeV/m for possible gradient field of the spacetime torsion. The
interferometer using atoms in $m_F=\pm 1$ is highly sensitive to the magnetic
field inhomogeneity, and a double differential measurement method is developed
to alleviate the inhomogeneity influence. Moreover, a proof experiment by
modulating the magnetic field is performed, which validates the alleviation of
the inhomogeneity influence in our test.
