We present a novel implementation strategy for distributed autoregressive moving average (ARMA) graph filters. Differently from the state of the art implementation, the proposed approach has the following benefits: (i) the designed filter coefficients come with stability guarantees, (ii) the linear convergence time can now be controlled by the filter coefficients, and (iii) the stable filter coefficients that approximate a desired frequency response are optimal in a least squares sense. Numerical results show that the proposed implementation outperforms the state of the art distributed infinite impulse response (IIR) graph filters. Further, even at fixed distributed costs, compared with the popular finite impulse response (FIR) filters, at high orders our method achieves tighter low-pass responses, suggesting that it should be preferable in accuracy-demanding applications.