In this paper we formulate and solve a worst-case system identification problem for single-input, single-output, linear, shift-invariant, distributed parameter plants. The available a prior information in this problem consists of time-dependent upper and lower bounds on the plant impulse response and the additive output noise. The available a posteriori information consists of a corrupt finite output time series obtained in response to a known, nonzero, but otherwise arbitrary, input signal. We present a novel identification method for this problem. This method maps the available a priori and a posteriori information into an "uncertain model" of the plant, which is comprised of a nominal plant model, a bounded additive output noise, and a bounded additive model uncertainty. The upper bound on the model uncertainty is explicit and expressed in terms of both the l1 and Hinfinity system norms. The identification method and the nominal model possess certain well-defined optimality properties and are computationally simple, requiring only the solution of a single linear programming problem.