We examine the effects of thermohaline mixing on the composition of the envelopes of low-metallicity asymptotic giant branch (AGB) stars. We have evolved models of 1, 1.5 and 2M(circle dot) from the pre-main sequence to the end of the thermally pulsing AGB with thermohaline mixing applied throughout the simulations. In agreement with other authors, we find that thermohaline mixing substantially reduces the abundance of He-3 on the upper part of the red giant branch in our lowest mass model. However, the small amount of He-3 that remains is enough to drive thermohaline mixing on the AGB. We find that thermohaline mixing is most efficient in the early thermal pulses and its efficiency drops from pulse to pulse. Nitrogen is not substantially affected by the process, but we do see substantial changes in C-13. The C-12/C-13 ratio is substantially lowered during the early thermal pulses, but the efficacy of the process is seen to diminish rapidly. As the process stops after a few pulses, the C-12/C-13 ratio is still able to reach values of 10(3)-10(4), which is inconsistent with the values measured in carbon-enhanced metal-poor stars. We also note a surprising increase in the Li-7 abundance, with log(10) epsilon(Li-7) reaching values of over 2.5 in the 1.5M(circle dot) model. It is thus possible to get stars which are both C and Li rich at the same time. We compare our models to measurements of carbon and lithium in carbon-enhanced metal-poor stars which have not yet reached the giant branch. These models can simultaneously reproduce the observed C and Li abundances of carbon-enhanced metal-poor turn-off stars that are Li rich, but the observed nitrogen abundances still cannot be matched.