Space-borne gravitational-wave interferometers such as LISA will detect the gravitational wave (GW) signal from the inspiral, plunge, and ringdown phases of massive black hole binary mergers at cosmological distances. From the inspiral waves, we will be able to measure the masses of the binaries' members; from the ringdown waves, we will be able to measure the mass of the final merged remnant. A subset of detected events allow the identification of both the inspiral and the ringdown waveforms in a given source and thus allow a measurement of the total mass-energy lost to GWs over the coalescence, M_GW. We define "golden" binary mergers to be those with measurement errors likely to be small enough for a physically useful determination of M_GW. A detailed sensitivity study, combined with simple black hole population models, suggests that a few golden binary mergers may be detected during a 3 yr LISA mission lifetime. Any such mass deficit measurement would constitute a robust and valuable observational test of strong-field relativistic gravity. An extension of this concept to include spin measurements may allow a direct empirical test of the black hole area theorem.