We present Fast Fourier Color Constancy (FFCC), a novel color constancy algorithm which works by reformulating the problem of illuminant estimation into a spatial localization task on a torus. On standard benchmarks, our model produces lower error rates than the previous state-of-the-art by $10-12\%$, while also being $250-3000\times$ faster. This speed and accuracy is primarily due to how FFCC primarily operates in the frequency domain, though this approach also introduces a set of new difficulties regarding aliasing, directional statistics and preconditioning, which we address. Unlike past work, our model produces a complete posterior distribution over illuminants instead of a single illuminant estimate, which allows for a richer analysis and enables a novel temporal smoothing technique. FFCC is capable of running at $\sim 700$ frames per second on a mobile phone, making it a viable solution to the problem of constructing an effective, real-time, temporally-coherent automatic white balance algorithm.