A search of neutrino magnetic moments was carried out at the Kuo-Sheng nuclear power station at a distance of 28 m from the 2.9 GW reactor core. With a high purity germanium detector of mass 1.06 kg surrounded by scintillating NaI(Tl) and CsI(Tl) crystals as anti-Compton detectors, a detection threshold of 5 keV and a background level of 1 kg(-1) keV(-1) day(-1) near threshold were achieved. Details of the reactor neutrino source, experimental hardware, background understanding, and analysis methods are presented. Based on 570.7 and 127.8 days of Reactor ON and OFF data, respectively, at an average Reactor ON electron antineutrino flux of 6.4x10(12) cm(-2) s(-1), the limit on the neutrino magnetic moments of mu(nu e)< 7.4x10(-11)mu(B) at 90% confidence level was derived. Indirect bounds on the nu(e) radiative decay lifetimes were inferred.