The time-integrated luminosity and average energy of the neutrino emission spectrum are essential diagnostics of core-collapse supernovae. The SN 1987A electron antineutrino observations by the Kamiokande-II and IMB detectors are only roughly consistent with each other and theory. Using new measurements of the star formation rate history, we reinterpret the Super-Kamiokande upper bound on the electron antineutrino flux from all past supernovae as an excluded region in neutrino emission parameter space. A gadolinium-enhanced Super-Kamiokande should be able to jointly measure these parameters, and a future megaton-scale detector would enable precision studies.PACS numbers: 97.60. Bw, 98.70.Vc, 95.85.Ry, 14.60.Pq When a massive star dies, its core collapses and rebounds, producing an outgoing shock wave that should eject the stellar envelope, causing the optical supernova, and leaving behind a neutron star remnant. However, in simulations, the shock wave stalls, leading to the whole star collapsing into a black hole, failing to produce an optical supernova or spread its heavy-element yields [1]. Since the required explosion energy is only ∼ 1% of the emergent neutrino energy, a full accounting of the neutrino emission is essential for understanding supernovae. Further, in the Bethe-Wilson delayed explosion model, the neutrinos revive the shock [2]. Resolution of the supernova problem would also have profound implications for the history of stellar evolution and nucleosynthesis.The weak interactions of neutrinos, which allow them to reveal the dynamics deep within the exploding star, also make their detection challenging. The last nearby supernova, SN 1987A, occurred in the Large Magellanic Cloud at 50 kpc, and ≃ 20 neutrinos were detected [3] preceding the optical supernova, confirming our basic understanding of the explosion [4]. However, even taking into account the small statistics, the fitted ranges for the time-integrated luminosity and average energy are perplexing, showing clear discrepancies among the experimental detections and theory [5,6]. A Milky Way supernova would yield many events in present detectors, but the expected supernova rate is only ∼ 3 per century. We have shown that with proposed megaton-scale detectors, it will be possible to build up the spectrum by detecting neutrinos one or two at a time from supernovae within 10 Mpc, at a rate as large as ∼ 1 neutrino per year [7].Here we propose a new approach, which could begin immediately, if the existing Super-Kamiokande (SK) detector were modified by the addition of gadolinium to greatly reduce backgrounds, as proposed by Beacom and Vagins [8,9]. We consider the spectrum of the Diffuse Supernova Neutrino Background (DSNB) [10,11,12,13] as the observable. The DSNB predictions depend on the redshift evolution of the supernova rate, which is separately measurable and increasingly well known, and the neutrino emission per supernova, the object of our study. While the received neutrino spectrum will be redshifted, it will have relatively high stat...