Cold dark matter particles with an intrinsic matter-antimatter asymmetry do not annihilate after gravitational capture by the Sun and can affect its interior structure. The rate of capture is exponentially enhanced when such particles have self-interactions of the right order to explain structure formation on galactic scales. A `dark baryon' of mass 5 GeV is a natural candidate and has the required relic abundance if its asymmetry is similar to that of ordinary baryons. We show that such particles can solve the `solar composition problem'. The predicted small decrease in the low energy neutrino fluxes may be measurable by the Borexino and SNO+ experiments.