We present an improved measurement of the carbon-nitrogen-oxygen (CNO) solar neutrino interaction rate at Earth obtained with the complete Borexino Phase-III dataset. The measured rate, R_{CNO}=6.7_{-0.8}^{+2.0} counts/(day×100 tonnes), allows us to exclude the absence of the CNO signal with about 7σ C.L. The correspondent CNO neutrino flux is 6.6_{-0.9}^{+2.0}×10^{8} cm^{-2} s^{-1}, taking into account the neutrino flavor conversion. We use the new CNO measurement to evaluate the C and N abundances in the Sun with respect to the H abundance for the first time with solar neutrinos. Our result of N_{CN}=(5.78_{-1.00}^{+1.86})×10^{-4} displays a ∼2σ tension with the "low-metallicity" spectroscopic photospheric measurements. Furthermore, our result used together with the ^{7}Be and ^{8}B solar neutrino fluxes, also measured by Borexino, permits us to disfavor at 3.1σ C.L. the "low-metallicity" standard solar model B16-AGSS09met as an alternative to the "high-metallicity" standard solar model B16-GS98.