Abstract In a neutron lifetime measurement at the Japan Proton Accelerator Complex, the neutron lifetime is calculated from the neutron decay rate and the incident neutron flux. The flux is obtained by counting the protons emitted from the neutron absorption reaction of ^{3}{\rm He}$ gas, which is diluted in a mixture of working gas in a detector. Hence, it is crucial to determine the amount of ^{3}{\rm He}$ in the mixture. In order to improve the accuracy of the number density of the ^{3}{\rm He}$ nuclei, we have suggested using the ^{14}{\rm N}({\rm n},{\rm p}){}^{14}{\rm C}$ reaction as a reference because this reaction involves similar kinetic energy to the $^3$He(n,p)$^3$H reaction and a smaller reaction cross section to introduce reasonable large partial pressure. The uncertainty of the recommended value of the cross section, however, is not satisfied with our requirement. In this paper we report the most accurate experimental value of the cross section of the $^{14}$N(n,p)$^{14}$C reaction at a neutron velocity of 2200 m s$^{-1}$, measured relative to the $^3$He(n,p)$^3$H reaction. The result was 1.868 $\pm$ 0.003 (stat.) $\pm$ 0.006 (sys.) b. Additionally, the cross section of the $^{17}$O(n,$\alpha$)$^{14}$C reaction at the neutron velocity is also redetermined as 249 $\pm$ 6 mb.