We studied on the magnetic damping constants (alpha) for L1(0)-FeNi and disordered FeNi employing three kinds of measurement methods. An L1(0)-FeNi thin film exhibited high perpendicular magnetic anisotropy of 7.1 x 10(6) erg cm(-3). At magnetic fields (H) lower than 2 kOe, alpha was estimated to be 0.091 +/- 0.003. However, it was reduced down to 0.013 +/- 0.001 with H, indicating that extrinsic contributions enhance alpha. The intrinsic alpha = 0.013 +/- 0.001 was comparable to alpha = 0.009 +/- 0.002 for the disordered FeNi. This suggests that L1(0)-FeNi is a candidate achieving high magnetic anisotropy and low magnetization damping simultaneously. (C) 2013 AIP Publishing LLC.