Electrons in a Dirac semimetals possess linear dispersion in all three spatial dimensions, and form part of a developing platform of novel quantum materials. Bi1-xSbx supports a three-dimensional Dirac cone at the Sb-induced band inversion point. Nanoscale phase-sensitive junction technology is used to induce superconductivity in this Dirac semimetal and to reveal its topological nature by the observation of Majorana zero modes in Nb-Bi0.97Sb0.03-Nb Josephson junctions. These zero modes are detected as a 4-periodic contribution to the current-phase relation under radio frequency irradiation. The conditions for a substantial Majorana contribution to the supercurrent are favourable because of the Dirac cone’s topological protection against backscattering and the limited number of modes in the junction due to the Dirac point lying within 20 meV of the Fermi level.