Three-dimensional (3D) Dirac semimetals are new quantum materials and can be viewed as 3D analogues of graphene. Many fascinating electronic properties have been proposed and realized in 3D Dirac semimetals, which demonstrate their potential applications in next generation quantum devices. Bismuth-antimony Bi1− x Sb x can be tuned from a topological insulator to a band insulator through a quantum critical point at x ≈ 4%, where 3D Dirac fermions appear. Here, we report on a magnetotransport study of Bi1− x Sb x at such a quantum critical point. An unusual magnetic-field induced semimetal-semiconductor phase transition was observed in the Bi0.96Sb0.04 single crystals. In a magnetic field of 8 T, Bi0.96Sb0.04 single crystals show giant magnetoresistances of up to 6000% at low-temperature, 5 K, and 300% at room-temperature, 300 K. The observed magnetoresistances keep linear down to approximate zero-field when the temperature is below 200 K. Our experimental results are not only interesting for the fundamental physics of 3D Dirac semimetals but also for potential applications of 3D Dirac semimetals in magnetoelectronic devices.