A theoretical study has been made of the effect of piezoelectric coupling on exceptional bulk waves (EBWs) that exist on the surface of a medium if the piezomoduli are made equal to zero. The analysis has been carried out assuming that the medium is of generic symmetry and that the piezoeffect is weak. In combination with elastic anisotropy, the piezoelectric coupling can cause the appearance of surface and leaky acoustic waves (SAWs and LSAWs) at orientations close to those supporting EBWs. The existence conditions for SAWs and LSAWs have been studied. In particular, it has been shown that the piezoeffect provides the existence of non–attenuating fast SAWs near the EBW associated with the branch of fast quasi–transverse bulk waves. Such an SAW appears independently of the crystallographic symmetry and material constants if the surface of the substrate is metallized. On the non–metallized surface, it may or may not exist, depending on the material constants. In addition, contrary to the case of non–piezoelectric media, quasi–longitudinal EBWs generally give rise to the leaky waves. Numerical computations for SiO2 crystal confirm our analytic estimations.