  We describe the first spectroscopic tomographic (spectrotomographic) weak
lensing measurement for a galaxy cluster based only on background galaxies with
spectroscopically determined redshifts. We use the massive cluster A2029 to
demonstrate the power of combining spectroscopy and lensing to obtain accurate
masses and to overcome biases from contamination and photometric redshift
errors. We detect the shear signal from the cluster at $>3.9 \sigma$. The shear
signal scales with source redshift in a way that is consistent with the angular
diameter distance ratio variation in a $\Lambda$CDM Universe. Furthermore, the
amplitude of the measured signal is consistent with the X-ray mass. Upcoming
spectroscopic instruments such as the Prime Focus Spectrograph on Subaru will
permit spectrotomographic weak lensing measurements with S/N comparable to
current photometric-redshift-based weak lensing measurements for hundreds of
galaxy clusters. Thus, spectrotomography may enable sensitive cosmological
constraints that complement and are independent of other measurement
techniques.
