We present in this work a new calculation of the standard-model benchmark value for the effective number of neutrinos, Neff-SM, that quantifies the cosmological neutrino-to-photon energy densities. The calculation takes into account neutrino flavour oscillations, finite-temperature effects in the quantum electrodynamics plasma to O(e3), where e is the elementary electric charge, and a full evaluation of the neutrino-neutrino collision integral. We provide furthermore a detailed assessment of the uncertainties in the benchmark Neff-SM value, through testing the value's dependence on (i) optional approximate modelling of the weak collision integrals, (ii) measurement errors in the physical parameters of the weak sector, and (iii) numerical convergence, particularly in relation to momentum discretisation. Our new, recommended standard-model benchmark is Neff-SM = 3.0440 ± 0.0002, where the nominal uncertainty is attributed predominantly to errors incurred in the numerical solution procedure (|δNeff| ∼10^-4), augmented by measurement errors in the solar mixing angle sin^2(θ12) (|δ eff| ∼10^-4).