  In a neutron lifetime measurement at the Japan Proton Accelerator Complex,
the neutron lifetime is calculated by the neutron decay rate and the incident
neutron flux. The flux is obtained due to 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 suggested to use the
${}^{14}{\rm N}({\rm n},{\rm p}){}^{14}{\rm C}$ reaction as a reference because
this reaction involves similar kinetic energy as the ${}^{3}{\rm He}({\rm
n},{\rm p}){}^{3}{\rm H}$ 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 ${}^{14}{\rm N}({\rm n},{\rm p}){}^{14}{\rm C}$ reaction at a
neutron velocity of 2200 m/s, measured relative to the ${}^{3}{\rm He}({\rm
n},{\rm p}){}^{3}{\rm H}$ reaction. The result was 1.868 $\pm$ 0.003 (stat.)
$\pm$ 0.006 (sys.) b. Additionally, the cross section of the ${}^{17}{\rm
O}({\rm n},{\rm \alpha}){}^{14}{\rm C}$ reaction at the neutron velocity is
also redetermined as 249 $\pm$ 6 mb.
