Alloys of Cu with nominally 5- and 10-at. ppm Fe were studied by transmission-electron spin resonance (TESR) in the temperature range 2-40 K. The samples were strained in order to remove background signals which hamper analysis of TESR in relatively pure metal films. These measurements represent the first reported observation of electron spin resonance in a dilute magnetic alloy below its Kondo temperature. The relaxation rate of these alloys roughly scales with Fe concentration and exhibits a clear minimum at approximately 25 K. The contribution of the Fe impurities to the relaxation rate of the alloys is (7.7 ± 0.9) × 108/sec. at. ppm at 25K. The g value of the alloys did not deviate from the g value of pure Cu within the experimental scatter. We analyze our results in terms of coupled Bloch equations and discuss possible mechanisms for the inferred local-moment-lattice and conduction-electron-lattice relaxation rates.