  We present an analytical method for finding all the electronic eigenfunctions
and eigenvalues of the aperiodic Thue-Morse lattice. We prove that this system
supports only extended electronic states which is a very unusual behavior for
this class of systems, and so far as we know, this is the only example of a
quasiperiodic or aperiodic system in which critical or localized states are
totally absent in the spectrum. Interestingly we observe that the symmetry of
the lattice leads to the existence of degenerate eigenstates and all the
eigenvalues excepting the four global band edges are doubly degenerate. We show
exactly that the Landauer resistivity is zero for all the degenerate
eigenvalues and it scales as $\sim L^2$ ($L=$ system size) at the global band
edges. We also find that the localization length $\xi$ is always greater than
the system size.
