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The possibility of an electric quadrupole Kondo effect for a non-Kramers doublet on a uranium (U) ion in a cubic metallic host is demonstrated by model calculations showing a Kondo upturn in the resistivity, universal quenching of the quadrupolar moment, and a heavy electron anomaly in the electronic specific heat. With inclusion of excited crystal-field levels, some of the unusual magnetic response data in the heavy-electron superconductor UBe 13 may be understood. Structural phase transitions at unprecedented low temperatures may occur in U-based heavy-electron materials.