  A combined theoretical and experimental study reveals evidence for the dual
topological insulating character of the stoichiometric natural superlattice
phase $\mathrm{Bi_{1}Te_{1}}=\mathrm{[Bi_{2}]_{1}[Bi_{2}Te_{3}]_{2}}$, being a
stack of alternating Bi bilayers and two quintuple layers of
$\mathrm{Bi_{2}Te_{3}}$. We identify $\mathrm{Bi_{1}Te_{1}}$ by density
functional theory to exhibit a non trivial time-reversal symmetry-driven
character of $\mathbb{Z}_{2}=(0;001)$ and additionally a mirror-symmetry
induced mirror Chern number of $n_{{\cal M}}=-2$, which indicates that
$\mathrm{Bi_{1}Te_{1}}$ is both a weak topological insulator and a topological
crystalline insulator. The coexistence of the two phenomena preordains distinct
crystal planes to host topological surface states that are protected by the
respective symmetries. The surface perpendicular to the stacking direction is
the 'dark' surface of the weak topological insulator, while hosting
mirror-symmetry protected surface states along the $\bar{\Gamma\mathrm{M}}$
direction at non-time-reversal invariant momenta points. We confirm the
stacking sequence of our MBE-grown $\mathrm{Bi_{1}Te_{1}}$ thin films by X-ray
diffraction and transmission electron microscopy, and find indications of the
topological crystalline and weak topological character in the surface
electronic spin structure by spin- and angle-resolved photoemission
spectroscopy, which nicely match the results from density functional theory.
