The power spectrum of the archetypal fluctuating bistable system, the underdamped double-well Duffing oscillator, is investigated both experimentally, with use of an electronic circuit, and theoretically. The experiment confirms previous analytic results for the structure of the spectrum, including the existence of three distinct peaks within a certain parameter range. The theoty is extended to describe analytically the shape of the peak due to intrawell fluctuations for arbitrary noise intensities as well as certain other features of the spectrum. Good quantitative agreement of theory and experiment is demonstrated.