  We report the first results from a new approach for measuring the $^3$He
neutron incoherent scattering length $b_{i}$. $b_{i}$ is directly proportional
to the difference $\Delta b=b_{+}-b_{-}$ in the two low-energy s-wave
neutron-nucleus scattering amplitudes $b_{+}$ and $b_{-}$, corresponding to the
singlet $J=0$ and triplet $J=1$ states of the neutron-$^3$He interaction,
respectively. An accurate measurement of $b_{i}$ can help distinguish among
different models of three-nucleon interactions by comparison to {\it ab initio}
nuclear theory calculations. The neutron birefringence caused by $\Delta b$
results in neutron spin rotation around the nuclear polarization. We measured
$\Delta b$ using polarized neutron spin rotation and the transmission of
neutrons through a $^3$He gas target polarized in situ by spin-exchange optical
pumping. This brief test measurement, conducted at the FZ-J\"ulich neutron spin
echo spectrometer at the Heinz Maier Leibnitz Zentrum (MLZ), yielded $\Delta b
= [-5.27 \pm 0.05$ (stat.) $- 0.05$ (syst.)] fm. We argue that this method can
be improved in precision to resolve the discrepancies between two prior
measurements of $b_i$ which are dependent on the polarized absorption cross
section $\sigma_p$. Further with absolute $^{3}$He polarization via NMR (in a
properly-shaped cell) concurrent with accurate neutron transmission
measurements, $\sigma_p$ can be measured to obtain independent values of
$b_{+}$ and $b_{-}$.
