Runaway positrons can be produced in the presence of runaway electron avalanches in magnetized plasmas. In this Letter, we determine the positron distribution, the fraction of runaway positrons, and the parametric dependences of their synchrotron radiation spectrum. We show that the maximum production occurs around γ(e)≃30, where γ(e) is the Lorentz factor of the fast electrons. For an avalanching positron distribution typical of tokamak plasmas, the maximum of the synchrotron radiation spectrum should be around a micron. The radiated power and spectrum shape are sensitive to the plasma parameters. Apart from its intrinsic interest, detection of radiation from positrons could be a diagnostic tool to understand the properties of the medium they propagate through.