  We perform a measurement of the tune-out wavelength, $\lambda_{0}$, between
the $D_{1}$, $6^2S_{1/2}\rightarrow6^2P_{1/2}$, and $D_{2}$,
$6^2S_{1/2}\rightarrow6^2P_{3/2}$, transitions for $^{133}$Cs in the ground
hyperfine state $(F=3, m_{F}=+3)$. At $\lambda_{0}$, the frequency-dependent
scalar polarizability is zero leading to a zero scalar ac Stark shift. We
measure the polarizability as a function of wavelength using Kapitza-Dirac
scattering of a $^{133}$Cs Bose-Einstein condensate in a one-dimensional
optical lattice, and determine the tune-out wavelength to be
$\lambda_{0}=880.21790(40)_{\textrm{stat}}(8)_{\textrm{sys}}$ nm. From this
measurement we determine the ratio of reduced matrix elements to be
$\left|\left<{6P_{3/2}\| d\|6S_{1/2}}\right>\right|^{2}\left|\left<{6P_{1/2}\|
d\|6S_{1/2}}\right>\right|^{2}=1.9808(2)$. This represents an improvement of a
factor of 10 over previous results derived from excited-state lifetime
measurements. We use the present measurement as a benchmark test of
high-precision theory.
