  {\alpha}-Sn with a diamond-type crystal structure provides an ideal avenue to
investigate novel topological properties owing to its rich diagram of
topological phases and simple elemental material structure. Thus far, however,
realisation of high-quality {\alpha}-Sn remains a challenge, which limits our
understanding of its quantum transport properties and device applications.
Here, we present epitaxial growth of {\alpha}-Sn on InSb (001) with the highest
quality thus far and reveal that it is a topological Dirac semimetal (TDS) by
quantum transport investigations together with first-principles calculations.
We realise unprecedentedly high quantum mobilities of both the surface state
(30000 cm^2/Vs), which is ten times higher than the previously reported values,
and the bulk heavy-hole (HH) state (1700 cm^2/Vs), which has never been
obtained experimentally. These excellent features allow us, for the first time,
to quantitatively characterise the nontrivial interfacial and bulk band
structure of {\alpha}-Sn via Shubnikov-de Haas oscillations at various
temperatures and under various magnetic field directions. These results reveal
the existence of a topological surface state (TSS) and a bulk HH band, both
with nontrivial phase shifts, indicating that the TDS phase of {\alpha}-Sn is
established. Furthermore, we demonstrate a crossover from the TDS to a
two-dimensional topological insulator (2D-TI) and a subsequent phase transition
to a trivial insulator when varying the thickness of {\alpha}-Sn. Our work
indicates that {\alpha}-Sn is an excellent model system to study novel
topological phases and a prominent material candidate for topological devices.
