In this paper, we determine the growth rate of the exponential radiation amplification in the ion-channel laser, where a relativistic electron beam wiggles in a focusing ion channel that can be created in a wakefield accelerator. For the first time the radiation diffraction, which can limit the amplification, is taken into account. The electron beam quality requirements to obtain this amplification are also presented. It is shown that both the beam energy and wiggler parameter spreads should be limited. Two-dimensional and three-dimensional particle-in-cell simulations of the self-consistent ion-channel laser confirm our theoretical predictions.