Mapping of annuli into canonical form.

Find the error in the following argument which suggests that it is not possible to map conformally a topological annulus into a canonical annulus:

The Riemann mapping theorem implies that any topological disk can be mapped to the unit disk. A topological annulus can therefore be mapped into the inside of a unit disk l z l ::::: I, with the outer boundary mapping into l z l = I and the inner boundary mapping into some closed curve inside the disk. To map this into a canonical annulus we m ust make the inner curve round and centered at the origin, while preserving the boundary l z l = I. This req uires a map of the disk to itself. The mappings of a disk to itself have three real parameters, just as the mappings of IHI to itself. It is not possible to round a general curve using just three parameters, so it is impossible to map the annulus into canonical form.