[en] We have determined the γ-ray branching ratio Γ_γ/Γ_tot for the first excited state in ¹⁴O (E_x=5.17 MeV) to be (3.1±1.7)x10^-4, corresponding to a γ width of 12±7 eV. This value was determined by measuring the ratio of the ¹⁴N(p,n)¹⁴O₀ to ¹⁴N(p,n)¹⁴O₁ cross sections via neutron time of flight, and the ratio of ¹⁴N(p,n)¹⁴O₀ to ¹⁴N(p,n)¹⁴O₁(γ)¹⁴O₀ cross sections at the same center-of-mass energy by bombarding a hydrogen target with a ¹⁴N beam and kinematically distinguishing the ¹⁴O₀ recoils from the two reaction channels. The present value for Γ_γ is consistent with other measurements at the 2-σ level; a weighted mean of all measurements is used to calculate the stellar ¹³N(p,γ)¹⁴O reaction rate, and thereby determine the temperatures and densities at which the stellar carbon-nitrogen-oxygen (CNO) cycle is converted to the hot CNO cycle