  We propose that the noncentrosymmetric LiGaGe-type hexagonal $ABC$ crystal
SrHgPb realizes a new type of topological semimetal that hosts both Dirac and
Weyl points in momentum space. The symmetry-protected Dirac points arise due to
a band inversion and are located on the sixfold rotation $z$-axis, whereas the
six pairs of Weyl points related by sixfold symmetry are located on the
perpendicular $k_z=0$ plane. By studying the electronic structure as a function
of the buckling of the HgPb layer, which is the origin of inversion symmetry
breaking, we establish that the coexistence of Dirac and Weyl fermions defines
a phase separating two topologically distinct Dirac semimetals. These two Dirac
semimetals are distinguished by the $\mathbb{Z}_2$ index of the $k_z=0$ plane
and the corresponding presence or absence of 2D Dirac fermions on side
surfaces. We formalize our first-principles calculations by deriving and
studying a low-energy model Hamiltonian describing the Dirac-Weyl semimetal
phase. We conclude by proposing several other materials in the
non-centrosymmetric $ABC $ material class, in particular SrHgSn and CaHgSn, as
candidates for realizing the Dirac-Weyl semimetal.
