We consider well-posed linear infinite-dimensional systems, the outputs of which are sampled in a generalized sense using a suitable weighting function. Under certain natural assumptions on the system, the weighting function, and the sampling period, we show that there exists a generalized hold function such that unity sampled-data feedback renders the closed-loop system exponentially stable ( in the state-space sense) as well as L(2)-stable ( in the input-output sense). To illustrate our main result, we describe an application to a structurally damped Euler - Bernoulli beam.