  We consider the condensate wavefunction of a rapidly rotating two-component
Bose gas with an equal number of particles in each component. If the
interactions between like and unlike species are very similar (as occurs for
two hyperfine states of $^{87}$Rb or $^{23}$Na) we find that the two components
contain identical rectangular vortex lattices, where the unit cell has an
aspect ratio of $\sqrt{3}$, and one lattice is displaced to the center of the
unit cell of the other. Our results are based on an exact evaluation of the
vortex lattice energy in the large angular momentum (or quantum Hall) regime.
