We propose a message propagation scheme for numerically stable inference in Gaussian graphical models which can otherwise be susceptible to errors caused by finite numerical precision. We adapt square root algorithms, popular in Kalman filtering, to graphs with arbitrary topologies. The method consists of maintaining potentials and generating messages that involve the square root of precision matrices. Combining this with the machinery of the junction tree algorithm leads to an efficient and numerically stable algorithm. Experiments are presented to demonstrate the robustness of the method to numerical errors that can arise in complex learning and inference problems.