Carbon, nitrogen, and oxygen are the fourth, sixth, and third most abundant elements in the Sun. Their abundances remain hotly debated due to the so-called solar modelling problem that has persisted for almost $20$ years. We revisit this issue by presenting a homogeneous analysis of $408$ molecular lines across $12$ diagnostic groups, observed in the solar intensity spectrum. Using a realistic 3D radiative-hydrodynamic model solar photosphere and LTE (local thermodynamic equilibrium) line formation, we find $logepsilon_{C} = 8.47pm0.02$, $logepsilon_{N} = 7.89pm0.04$, and $logepsilon_{O} = 8.70pm0.04$. The stipulated uncertainties mainly reflect the sensitivity of the results to the model atmosphere; this sensitivity is correlated between the different diagnostic groups, which all agree with the mean result to within $0.03$ dex. For carbon and oxygen, the molecular results are in excellent agreement with our 3D non-LTE analyses of atomic lines. For nitrogen, however, the molecular indicators give a $0.12$ dex larger abundance than the atomic indicators, and our best estimate of the solar nitrogen abundance is given by the mean: $7.83$ dex. The solar oxygen abundance advocated here is close to our earlier determination of $8.69$ dex, and so the present results do not significantly alleviate the solar modelling problem.
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