Dirac and Weyl semimetals are 3D analogues of graphene in which crystalline symmetry protects the nodes against gap formation(1-3). Na3Bi and Cd3As2 were predicted to be Dirac semimetals(4,5), and recently confirmed to be so by photoemission experiments(6-8). Several novel transport properties in a magnetic field have been proposed for Dirac semimetals(2,9-11). Here, we report a property of Cd3As2 that was unpredicted, namely a remarkable protection mechanism that strongly suppresses backscattering in zero magnetic field. In single crystals, the protection results in ultrahigh mobility, 9 x 10(6) cm(2) V-1 s(-1) at 5 K. Suppression of backscattering results in a transport lifetime 10(4) times longer than the quantum lifetime. The lifting of this protection by the applied magnetic field leads to a very large magnetoresistance. We discuss how this may relate to changes to the Fermi surface induced by the applied magnetic field.