  ^{139}La NMR measurements of La_{2/3}Ca_{1/3}MnO_3 performed in the
paramagnetic state and high magnetic fields are reported. Analysis of the high
temperature spectrum measured establishes that the spectrum it is a standard
powder pattern broadened by a variation in lattice distortions. At lower
temperatures, the spectrum shifts and broadens. Both the shift and broadening
exhibit Curie-Weiss behavior, indicating that the shift measures the
polarization of the electron spin system, and the broadening reflects a
distribution of magnetic susceptibilities. This distribution may result from
variations of local susceptibility in the bulk of the sample or from
differences in demagnetizing factors among powder grains. The spectrum also
indicates that the lattice distortions do not change as the temperature lowers.
Spectral diffusion measurements suggest that the temperature dependence of the
spectrum shape does not result from freezing out of motion of magnetic
polarons. Variation of the nuclear spin-lattice relaxation across the spectrum
indicate that magnetic fluctuations, not lattice vibrations, dominate nuclear
relaxation. Nuclear spin-lattice relaxation therefore measures electron spin
dynamics in this system. The magnetic field dependence of the spin-lattice
relaxation indicates that the electron spin-spin correlation function adopts
simple single exponential behavior with a slow field-independent correlation
time of 10^{-8} seconds near Tc. The spin-spin correlation function changes
form at higher temperatures, but remains slow. The spin-lattice relaxation
therefore indicates the presence of extremely slow dynamics above Tc.
