We present the first investigation of the electrical properties (electron Hall mobilities and concentrations) of 1-mu m thick relaxed Ge layers grown by surfactant-mediated epitaxy (SME) with the surfactant Sb on Si(111) substrates at growth temperatures between 640 degrees C and 720 degrees C. We found that with rising growth temperatures the Ge layer quality improves as is demonstrated by the mobility increases at 300 K and 77 K. A record high electron mobility of 3100 cm(2)/Vs at 300 K (12 300 cm(2)/Vs at 77 K) which is close to the Ge lattice mobility was measured in the 720 degrees C layer together with an electron concentration of 1.1 x 10(16) cm(-3). This was derived from a differential Hall analysis and is identical to an interpolation value obtained from a comparison with high-qualify bulk Ge mobility data. A secondary ion mass spectroscopy (SIMS) analysis yielded a concentration below the detection limit of 2 x 10(17) cm(-3) Sb. Our results demonstrate an Sb background doping reduction of three orders of magnitude in SME-grown Ge/Si layers compared to earlier reports. We interpret this as being due to an enhanced surfactant segregation without kinetic limitations at high growth temperatures. The high quality, low doping and high mobility of these Ge layers suggests an exciting potential for future device applications.