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We report on investigations of magnetoresistance (MR) in polycrystalline La2/3Ca1/3Mn1-xCuxO3 (x = 0 and 4%) samples prepared by the sol-gel method followed a sintering treatment at 1000℃. For the x = 4% sample, the most remarkable MR effect appears at temperatures near 77K and for magnetic fields lower than 1 T. For fields ~ 0.3 T and at temperatures ~ 77 K, the x = 4% sample exhibits a giant MR effect with Mro (≡△p/p (H = 0)) between ~ 50 and ~ 70%, comparable to the so-called colossal MR response observed in the x = 0 sample for the 3 T field.