  Low-frequency spin dynamics in the S=1/2 antiferromagnetic spin-chain
compound \alpha-VO(PO_3)_2 has been studied by means of 31P NMR. The nuclear
spin-lattice relaxation rate 1/T_1 at the P site exhibits H^{-1/2} dependence
on the applied magnetic field (H) at temperatures (T) well above the intrachain
coupling strength J/k_{B} = 3.50 K indicating one-dimensional diffusive spin
dynamics. The diffusive contribution to 1/T_1 decreases on cooling as
electronic spins acquire short-range antiferromagnetic correlations within the
chain, and vanishes almost entirely around T \approx J/k_{B}. This is
accompanied by an apparent increase of the spin-diffusion constant from the
value expected in the classical limit. On the other hand, the field-independent
part of 1/T_1 increases with decreasing temperature, which may be a precursor
for the true long-range antiferromagnetic ordering found below T_{N} = 1.93 \pm
0.01 K.
