In a Dirac semimetal, each Dirac node is resolved into two Weyl nodes with opposite "handedness" or chirality. The two chiral populations do not mix. However, in parallel electric and magnetic fields (E parallel to B), charge is predicted to flow between the Weyl nodes, leading to negative magnetoresistance. This "axial" current is the chiral (Adler-Bell-Jackiw) anomaly investigated in quantum field theory. We report the observation of a large, negative longitudinal magnetoresistance in the Dirac semimetal Na3Bi. The negative magnetoresistance is acutely sensitive to deviations of the direction of B from E and is incompatible with conventional transport. By rotating E (as well as B), we show that it is consistent with the prediction of the chiral anomaly.