Confirming the origin of Gilbert damping by experiment has remained a challenge for many decades, even for simple ferromagnetic metals. In this Letter, we experimentally identify Gilbert damping that increases with decreasing electronic scattering in epitaxial thin films of pure Fe. This observation of conductivity-like damping, which cannot be accounted for by classical eddy current loss, is in excellent quantitative agreement with theoretical predictions of Gilbert damping due to intraband scattering. Our results resolve the longstanding question about a fundamental damping mechanism and offer hints for engineering low-loss magnetic metals for cryogenic spintronics and quantum devices.