Neutrons produced by the carbon fusion reaction 12C(12C,n)23Mg play an important role in stellar nucleosynthesis. 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 which extends deep into the astrophysical energy range along with a new and improved extrapolation technique based on experimental data from the mirror reaction 12C(12C,p)23Na. The new reaction rate has been determined with a well-defined uncertainty which exceeds the precision required by astrophysics models. Using our constrained rate, we find that 12C(12C,n)23Mg is crucial to the production of Na and Al in Pop-III Pair Instability Supernovae.