  We present a feasibility study of a simultaneous sub-percent extraction of
the weak charge and the weak radius of the ${}^{12}$C nucleus using
parity-violating electron scattering, based on a largely model-independent
assessment of the uncertainties. The corresponding measurement is considered to
be carried out at the future MESA facility in Mainz with $E_{\rm beam} = 155$
MeV. We find that a combination of a $0.3\%$ precise measurement of the
parity-violating asymmetry at forward angles with a $10\%$ measurement at
backward angles will allow to determine the weak charge and the weak radius of
${}^{12}$C with $0.4\%$ and $0.5\%$ precision, respectively. These values could
be improved to $0.3\%$ and $0.2\%$ for a $3\%$ backward measurement. This
experimental program will have impact on precision low-energy tests in the
electroweak sector and nuclear structure.
