  Photoemission experiments have shown that Bi$_2$Se$_3$ is a topological
insulator. By controlled doping, we have obtained crystals of Bi$_2$Se$_3$ with
non-metallic conduction. At low temperatures, we uncover a novel type of
magnetofingerprint signal which involves the spin degrees of freedom. Given the
mm-sized crystals, the observed amplitude is 200-500$\times$ larger than
expected from universal conductance fluctuations. The results point to very
long phase breaking lengths in an unusual conductance channel in these
non-metallic samples. We discuss the nature of the in-gap conducting states and
their relation to the topological surface states.
