We investigate the magnetic anisotropy of epitaxial Fe films on GaN(0001) and GaN(0001). Due to the particular orientation relationship between Fe and GaN{0001}, the leading term proportional to K 1 does not contribute to the magnetic anisotropy, allowing us to determine K 3 with unprecedented accuracy. Using two experimental techniques, we obtain values for K 3 of 4.7-9.2 × 10 5 erg/cm 3 , i.e., comparable to and even larger than K 1 .