  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.
