Ionic conduction by H+ and D+ in doped perovskites BaCe(1-x)M(x)O3-alpha (M = Y, Gd; x = 0.05, 0.10) at high temperatures in H2O- and D2O-moistened atmospheres is investigated. Conduction by hydrogen ions was confirmed by emf measurements on various gas cells using specimen ceramics as the solid electrolyte membrane separating moist and 'dry' nitrogen atmospheres. Ionic charge carrier densities in these materials are very low, as demonstrated by thermogravimetric and infrared measurements. Electrical conductivity (sigma) measurements were carried out in the temperature range T = 600-900-degrees-C using complex impedance techniques. Plots of log sigma-T versus 1/T showed Arrhenius behaviour in all cases. Attempt frequencies for charge-carrier migration correspond to deformation of the Ce/M-O-H bond angle, with H+ migrating between sites corresponding to attachment to neighbouring oxide ions. An isotope effect is seen in the activation barriers, E(a) (ca. 33 kJ mol-1 for H+ and ca. 43 kJ mol-1 for D+). This results from tunnelling of charge carriers through a time-dependent barrier arising from coupling to lattice phonons.