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We study neutrino emission from the remnant of an inspiraling binary neutronstar following coalescence. The mass of the merged remnant is likely to exceedthe stability limit of a cold, rotating neutron star. However, the angularmomentum of the remnant may also approach or even exceed the Kerr limit, J/M^2= 1, so that total collapse may not be possible unless some angular momentum isdissipated. We find that neutrino emission is very inefficient in decreasingthe angular momentum of these merged objects and may even lead to a smallincrease in J/M^2. We illustrate these findings with a post-Newtonian,ellipsoidal model calculation. Simple arguments suggest that the remnant mayform a bar mode instability on a timescale similar to or shorter than theneutrino emission timescale, in which case the evolution of the remnant will bedominated by the emission of gravitational waves.Comment: 12 pages AASTeX, 2 figures, to appear in Ap