The aim of this paper is to analyze the delay--dependent stability of symmetric Runge--Kutta methods, including the Gauss methods and the Lobatto IIIA, IIIB and IIIS methods, for the linear neutral delay integro--differential equations. By means of the root locus technique, the structure of the root locus curve is given and the numerical stability region of symmetric Runge--Kutta methods is obtained. It is proved that, under some conditions, the analytical stability region is contained in the numerical stability region. Finally, some numerical examples are presented to illustrate the theoretical results.