The / isotopic ratio is a widely used tracer of nucleosynthesis and mixing processes in red giant stars. Its value depends on the rates of the proton-capture reactions responsible for the production and destruction of and among which the reaction plays a central role, as it involves both isotopes. This work reports on a new measurement of the cross section carried out at the Laboratory for Underground Nuclear Astrophysics (LUNA) over a center-of-mass energy range of 200–380~keV, a region previously unexplored by experimental studies. Two complementary techniques were employed: prompt gamma-ray detection and activation. The results obtained with the two techniques are consistent within experimental uncertainties. From the measured cross sections and existing literature data, we evaluate a new thermonuclear reaction rate, which is then used to demonstrate how simultaneous measurements of stellar carbon and oxygen isotopic ratios constrain the initial mass and the extent of the mixing events in red giant branch stars. In particular, we demonstrate how the ${}^{12}$C/${}^{13}$C abundance ratio is primarily sensitive to the temperature T${}_{\mathrm{\text{max}}}$ attained during mixing, whereas the ${}^{16}$O/${}^{17}$O abundance ratio is highly sensitive to the initial stellar mass M${}_i$. Therefore, simultaneous measurements of these isotopic ratios can disentangle and place stringent constraints on M${}_i$ and T${}_{\mathrm{\text{max}}}$, and the remaining uncertainty is now dominated by stellar abundance measurements rather than nuclear-physics inputs.