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Understanding the magnetic excitations in high-temperature (high-T-c) copper-oxide superconductors is important because they may mediate the electron pairing for superconductivity(1,2). By determining the wavevector (Q) and energy ((h) over bar omega) dependence of the magnetic excitations, it is possible to calculate the change in the exchange energy available to the superconducting condensation energy(3-5). For the high-T-c superconductor YBa2Cu3O6+x, the most prominent feature in the magnetic excitations is the resonance(6-12). Suggestions that the resonance contributes a major part of the superconducting condensation(4,13) have not gained acceptance because the resonance is only a small portion of the total magnetic scattering(12-14). Here, we report an extensive mapping of magnetic excitations for YBa2Cu3O6.95 (T-c similar to 93 K). Absolute intensity measurements of the full spectra allow us to estimate the change in the magnetic exchange energy between the normal and superconducting states, which is about 15 times larger than the superconducting condensation energy(15,16) - more than enough to provide the driving force for high-T-c superconductivity in YBa2Cu3O6.95.