The roughness of tetrahedral amorphous carbon (ta-C) films grown at room temperature is measured as a function of film thickness by atomic force microscopy, to extract roughness and growth exponents of alpha approximately 0.39 and beta approximately 0-0.1, respectively. This extremely small growth exponent shows that some form of surface diffusion and relaxation operates at a homologous temperature of 0.07, much lower than in any other material. This is accounted for by a Monte Carlo simulation, which assumes a smoothening during a thermal spike, following energetic ion deposition. The low roughness allows ta-C to be used as an ultrathin protective coating on magnetic disk storage systems with approximately 1 Tbit/in.(2) storage density.