The effect of a homogeneous anisotropic initial stress on the propagation of infinitesimal amplitude elastic body waves in a perfectly elastic, homogeneous medium is investigated. If the medium is inherently isotropic in the reference configuration and if the magnitude τ0 of the deviatoric part of the initial static stress is small compared to the rigidity μ of the medium, then the apparent body-wave velocities of P waves are unaffected by the initial stress to first order in τ0/μ. The apparent body-wave velocities of S waves are rendered anisotropic to first order, and this effect is described explicitly. It is concluded that the direct effect of an anisotropic initial stress cannot contribute appreciably to the observed velocity anisotropy of horizontally propagating P waves in the oceanic upper mantle. Those observations require an inherent elastic anisotropy of the oceanic uppermantle material.