Cooper pairs in non-centrosymmetric superconductors can acquire finite centre-of-mass momentum in the presence of an external magnetic field. Recent theory predicts that such finite-momentum pairing can lead to an asymmetric critical current, where a dissipationless supercurrent can flow along one direction but not in the opposite one. Here we report the discovery of a giant Josephson diode effect in Josephson junctions formed from a type-II Dirac semimetal, NiTe₂. A distinguishing feature is that the asymmetry in the critical current depends sensitively on the magnitude and direction of an applied magnetic field and achieves its maximum value when the magnetic field is perpendicular to the current and is of the order of just 10 mT. Moreover, the asymmetry changes sign several times with an increasing field. These characteristic features are accounted for by a model based on finite-momentum Cooper pairing that largely originates from the Zeeman shift of spin-helical topological surface states. The finite pairing momentum is further established, and its value determined, from the evolution of the interference pattern under an in-plane magnetic field. The observed giant magnitude of the asymmetry in critical current and the clear exposition of its underlying mechanism paves the way to build novel superconducting computing devices using the Josephson diode effect.