Dirac semimetals and Weyl semimetals are 3D analogs 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 [6-8]. Several novel transport properties in a magnetic field H have been proposed for Dirac semimetals [2, 10, 11, 16]. Here we report an interesting property in Cd3As2 that was unpredicted, namely a remarkable protection mechanism that strongly suppresses back-scattering in zero H. In single crystals, the protection results in ultrahigh mobility, 9 × 10 6 cm 2 /Vs at 5 K. Suppression of backscattering results in a transport lifetime 10 4 × longer than the quantum lifetime. The lifting of this protection by H leads to a very large magnetoresistance. We discuss how this may relate to changes to the Fermi surface induced by H.