  Using the model independent average neutron rms radius of $^{133}\text{Cs}$
and $^{127}\text{I}$ obtained from the analysis of the coherent elastic
neutrino-nucleus scattering data of the COHERENT experiment, we remove the
long-standing $1.5\,\sigma$ tension between the Standard Model prediction and
the weak mixing angle measurement from the atomic parity violation (APV) in
caesium. The updated APV result becomes $\sin^2
\vartheta_{\textrm{W}}=0.239{}^{+0.006}_{-0.007}$, to be compared with the
Standard Model prediction at low momentum transfer, $\sin^2
\vartheta_{\text{W}}^{\textrm{SM}} = 0.23857(5)$. Moreover, exploiting the fact
that the APV result is highly sensitive to the caesium neutron rms radius,
$R_{n}$, and assuming that the Standard Model is correct, we combine the APV
and the COHERENT measurements in order to get a better determination of
$R_{n}$. The value of $R_{n}=5.42\pm 0.31\,\textrm{fm}$ is obtained, improving
significantly the current uncertainty. This result allows to infer a meaningful
value of the caesium neutron skin, the difference between the neutron and
proton distribution radii, equal to $\Delta R_{np}=0.62\pm 0.31\,\text{fm}$,
showing for the first time a $2\, \sigma$ deviation from zero.
