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We consider the identification of a single open crack in a simply supported beam having nonuniform smooth profile and undergoing infinitesimal in-plane flexural vibration. The profile is assumed to be symmetric with respect to the mid-point of the beam axis. The crack is modeled by inserting a rotational linearly elastic spring at the damaged cross-section. We establish sufficient conditions for the unique identification of the crack by a suitable pair of natural frequency data, and we present a constructive algorithm for determining the damage parameters. The result is proved under a technical a priori assumption on the zeros of a suitable function determined in terms of the eigenfunctions of the problem. Extensions to beams under different sets of end conditions are also discussed. Theoretical results are confirmed by an extensive numerical investigation, both on simulated and experimental data.