Incoherent quasi-elastic neutron scattering spectra have been measured down to low scattering vectors (Q >) 0.14 Å-1) on magnetically aligned specimens of the nematic and smectic A phases of EABAC. Data were obtained for Q || n and Q⊥n where n denotes the direction of the unique axis) giving the diffusion coefficients D|| and D⊥ for the two phases as follows (in units of 10-7 cm2 s-1); nematic: D|| = 15.8 ± 2.0, D⊥ = 12.3 ± 0.8; smectic A: D|| = 4.3 ± 0.4, D⊥ = 4.9 ± 0.8. The anisotropy is reversed between the phases and the molecules are more mobile in the nematic phase as expected. In the smectic A experiments with Q || n an apparent inconsistency between low and high Q results is interpreted as evidence for the existence of some slow motion other than translational diffusion which requires further investigation. Preliminary measurements were made to explore the application of the "fixed window" method for determining the temperature dependence of the diffusion coefficients.