The ^{12}C/^{13}C ratio is a significant indicator of nucleosynthesis and mixing processes during hydrogen burning in stars. Its value mainly depends on the relative rates of the ^{12}C(p,γ)^{13}N and ^{13}C(p,γ)^{14}N reactions. Both reactions have been studied at the Laboratory for Underground Nuclear Astrophysics (LUNA) in Italy down to the lowest energies to date (E_{c.m.}=60 keV) reaching for the first time the high energy tail of hydrogen burning in the shell of giant stars. Our cross sections, obtained with both prompt γ-ray detection and activation measurements, are the most precise to date with overall systematic uncertainties of 7%-8%. Compared with most of the literature, our results are systematically lower, by 25% for the ^{12}C(p,γ)^{13}N reaction and by 30% for ^{13}C(p,γ)^{14}N. We provide the most precise value up to now of 3.6±0.4 in the 20-140 MK range for the lowest possible ^{12}C/^{13}C ratio that can be produced during H burning in giant stars.