We have observed the conduction electron spin resonance (CESR) in fine powders of MgB2 both in the superconducting and normal states. The Pauli susceptibility is chi_s=2.0*10^{-5} emu/mole in the temperature range of 450 to 600 K. The spin relaxation rate has an anomalous temperature dependence. The CESR measured below T_c at several frequencies suggests that MgB_2 is a strongly anisotropic superconductor with the upper critical field, H_c2, ranging between 2 and 16 T. The high-field reversible magnetization data of a randomly oriented powder sample are well described assuming that MgB_2 is an anisotropic superconductor with H_c2^{ab} / H_{c2}^{c} approx 6--9.
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