  The complete gauge-invariant set of the one-loop QED corrections to the
parity-nonconserving (PNC) amplitude in cesium and francium is evaluated to all
orders in $\alpha Z$ using a local form of the Dirac-Fock potential. The
calculations are performed in both length and velocity gauges for the absorbed
photon and the total binding QED correction is found to be $-$0.27(3)% for Cs
and $-$0.28(5)% for Fr. Moreover, a high-precision calculation of the
electron-correlation and Breit-interaction effects on the 7$s-8s$ PNC amplitude
in francium using a large-scale configuration-interaction Dirac-Fock method is
performed. The obtained results are employed to improve the theoretical
predictions for the PNC transition amplitude in Cs and Fr. Using an average
value from two most accurate measurements of the vector transition
polarizability, the weak charge of $^{133}$Cs is derived to amount to $
Q_W=-72.65(29)_{\rm exp}(36)_{\rm theor}$. This value deviates by $1.1\sigma$
from the prediction of the standard model. The values of the $7s$-$8s$ PNC
amplitude in $^{223}$Fr and $^{210}$Fr are obtained to be $-$15.49(15) and
$-$14.16(14), respectively, in units of i$\times 10^{-11}(-Q_W)/N$ a.u.
