The complete gauge-invariant set of the one-loop QED corrections to the parity-nonconserving 6s-7s amplitude in 133Cs is evaluated to all orders in alphaZ using a local version of the Dirac-Hartree-Fock potential. The calculations are performed in both length and velocity gauges for the absorbed photon. The total binding QED correction is found to be -0.27(3)%. The weak charge of 133Cs, derived using two most accurate values of the vector transition polarizability beta, is Q(W)=-72.57(46) for beta=26.957(51)a(3)(B) and Q(W)=-73.09(54) for beta=27.15(11)a(3)(B). The first value deviates by 1.1sigma from the prediction of the standard model, while the second one is in perfect agreement with it.