  We report a determination of the n-$^3$He scattering length difference
$\Delta b^{\prime} = b_{1}^{\prime}-b_{0}^{\prime} = $ ($-5.411$ $\pm$ $0.031$
(statistical) $\pm$ $0.039$ (systematic)) fm between the triplet and singlet
states using a neutron interferometer. This revises our previous result $\Delta
b^{\prime} = $ (-5.610 $\pm$ $0.027$ (statistical) $\pm$ $0.032$ (systematic)
fm obtained using the same technique in 2008. This revision is due to a
re-analysis of the 2008 experiment that includes a more robust treatment of the
phase shift caused by magnetic field gradients near the $^3$He cell.
Furthermore, we more than doubled our original data set from 2008 by acquiring
six months of additional data in 2013. Both the new data set and a re-analysis
of the older data are in good agreement. Scattering lengths of low Z isotopes
are valued for use in few-body nuclear effective field theories, provide
important tests of modern nuclear potential models and in the case of $^3$He
aid in the interpretation of neutron scattering from quantum liquids. The
difference $\Delta b^{\prime}$ was determined by measuring the relative phase
shift between two incident neutron polarizations caused by the spin-dependent
interaction with a polarized $^3$He target. The target $^3$He gas was sealed
inside a small, flat windowed glass cell that was placed in one beam path of
the interferometer. The relaxation of $^3$He polarization was monitored
continuously with neutron transmission measurements. The neutron polarization
and spin flipper efficiency were determined separately using $^3$He analyzers
and two different polarimetry analysis methods. A summary of the measured
scattering lengths for n-$^3$He with a comparison to nucleon interaction models
is given.
