We present in this work a new calculation of the standard-model benchmark value for the effective number of neutrinos, $N_{rm eff}^{rm SM}$, 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 ${cal O}(e^3)$, 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 $N_{rm eff}^{rm SM}$ value, through testing the values 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 $N_{rm eff}^{rm SM} = 3.0440 pm 0.0002$, where the nominal uncertainty is attributed predominantly to errors incurred in the numerical solution procedure ($|delta N_{rm eff}| sim10^{-4}$), augmented by measurement errors in the solar mixing angle $sin^2theta_{12}$ ($|delta N_{rm eff}| sim10^{-4}$).
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