It is shown that the problem of neoclassical transport at low collisionality in a perfectly quasi-isodynamic (or omnigeneous) stellarator can be reduced to the corresponding problem in a tokamak. In fact, the distribution function consists of two parts: one that can be calculated exactly (for any collision operator) and does not carry any net parallel current and one that is proportional to the total toroidal current and is determined by an equation that is identical to that solved in tokamak theory. Results from the latter can therefore be carried over to the corresponding stellarator situation. Specifically, if the total toroidal current enclosed by a flux surface vanishes, then the net bootstrap current on that surface also vanishes. It is therefore consistent to optimize a stellarator in such a way that the bootstrap current and neoclassical transport are simultaneously minimized.