In this paper, we focus on the problem of interpolating a continuous-time AR(1) process with stable innovations using minimum average error criterion. Stable innovations can be either Gaussian or non-Gaussian. In the former case, the optimality of the exponential splines is well understood. For non- Gaussian innovations, however, the problem has been all too often addressed through Monte Carlo methods. In this paper, based on a recent non-Gaussian stochastic framework, we revisit the AR(1) processes in the context of stable innovations and we derive explicit expressions for the optimal interpolator. We find that the interpolator depends on the stability index of the innovation and is linear for all stable laws, including the Gaussian case. We also show that the solution can be expressed in terms of exponential splines.