A small-gain approach is presented for analyzing exponential stability of a class of (dynamical) hybrid systems. The systems considered in the paper are composed of finite-dimensional dynamics, represented by a linear ordinary differential equation (ODE), and infinite-dimensional dynamics described by a parabolic partial differential equation (PDE). Exponential stability is established under conditions involving the maximum allowable sampling period (MASP). This new stability result is shown to be useful in the design of sampled-output exponentially convergent observers for linear systems that are described by an ODE-PDE cascade. The new stability result also proves to be useful in designing practical approximate observers involving no PDEs.