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We estimate the production rate of axion-type particles in the core of the Earth, at a temperature T approximate to 5000 K. We constrain thermal geo-axion emission by demanding a core-cooling rate less than O(100) K/Gyr, as suggested by geophysics. This yields a "nonstellar" (unaffected by extreme stellar temperatures or densities) bound on the axion-electron (ae) fine structure constant, alpha(ae)less than or similar to 10(-18), stronger than the existing accelerator (vacuum) bound by 4 orders of magnitude. We consider the prospects for measuring the geo-axion flux through conversion into photons in a geoscope; such measurements can further constrain alpha(ae).