This paper deals with the establishment of Input-to-State Stability (ISS) estimates for infinite dimensional systems with respect to both boundary and distributed disturbances. First, a new approach is developed for the establishment of ISS estimates for a class of Riesz-spectral boundary control systems satisfying certain eigenvalue constraints. Second, a concept of weak solutions is introduced in order to relax the disturbances regularity assumptions required to ensure the existence of classical solutions. The proposed concept of weak solutions, that applies to a large class of boundary control systems which is not limited to the Riesz-spectral ones, provides a natural extension of the concept of both classical and mild solutions. Assuming that an ISS estimate holds true for classical solutions, we show the existence, the uniqueness, and the ISS property of the weak solutions.