Neutrons produced by the carbon fusion reaction (12)C((12)C,n)(23)Mg play an important role in stellar nucleosynthesis. However, past studies have shown large discrepancies between experimental data and theory, leading to an uncertain cross section extrapolation at astrophysical energies. We present the first direct measurement that extends deep into the astrophysical energy range along with a new and improved extrapolation technique based on experimental data from the mirror reaction (12)C((12)C,p)(23)Na. The new reaction rate has been determined with a well-defined uncertainty that exceeds the precision required by astrophysics models. Using our constrained rate, we find that (12)C((12)C,n)(23)Mg is crucial to the production of Na and Al in pop-III pair instability supernovae. It also plays a nonnegligible role in the production of weak s-process elements, as well as in the production of the important galactic γ-ray emitter (60)Fe.