  Using the new results on coherent elastic neutrino-nucleus scattering data in
cesium-iodide provided by the COHERENT experiment, we determine a new
measurement of the average neutron rms radius of $^{133}\text{Cs}$ and
$^{127}\text{I}$. In combination with the atomic parity violation (APV)
experimental result, we derive the most precise measurement of the neutron rms
radii of $^{133}\text{Cs}$ and $^{127}\text{I}$, disentangling for the first
time the contributions of the two nuclei. By exploiting these measurements we
determine the corresponding neutron skin values for $^{133}\text{Cs}$ and
$^{127}\text{I}$. These results suggest a preference for models which predict
large neutron skin values, as corroborated by the only other electroweak
measurements of the neutron skin of $^{208}\text{Pb}$ performed by PREX
experiments. Moreover, for the first time, we obtain a data-driven APV+COHERENT
measurement of the low-energy weak mixing angle with a percent uncertainty,
independent of the value of the average neutron rms radius of $^{133}\text{Cs}$
and $^{127}\text{I}$, that is allowed to vary freely in the fit. The value of
the low-energy weak mixing angle that we found is slightly larger than the
standard model prediction.
