We report the first ^{75}As-NMR study on a single crystal of the hole-doped iron-pnictide superconductor Ba_{0.7}K_{0.3}Fe_2As_{2} (T_c = 31.5 K). We find that the Fe antiferromagnetic spin fluctuations are anisotropic and are weaker compared to underdoped copper-oxides or cobalt-oxide superconductors. The spin lattice relaxation rate 1/T_1 decreases below T_c with no coherence peak and shows a step-wise variation at low temperatures, which is indicative of multiple superconducting gaps, as in the electron-doped Pr(La)FeAsO$_{1-x}$F$_{x}$. Furthermore, no evidence was obtained for a microscopic coexistence of a long-range magnetic and superconductivity.
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