We investigate the problem of satellite survival during a hypothetical Late Heavy Bombardment in the outer solar system, as predicted by the Nice model (Tsiganis, Gomes, Morbidelli, and Levison 2005, Nature 435). Using a Monte Carlo approach we calculate, for satellites of Jupiter, Saturn, and Uranus, the probability of experiencing a catastrophic collision during the Late Heavy Bombardment (LHB). We find that Mimas, Enceladus, Tethys, and Miranda experience at least one catastrophic impact in every simulation. Because reaccretion is expected to be rapid, these bodies will have emerged as scrambled mixtures of rock and ice. Tidal heating may have subsequently modified the latter three, but in the nominal LHB model Mimas should be a largely undifferentiated, homogeneous body. A differentiated Mimas would imply either that this body formed late or that the Nice model requires significant modification.