  The {\it intrinsic} decoherence from vibrational coupling of the ions in the
Cirac-Zoller quantum computer [Phys. Rev. Lett. {\bf 74}, 4091 (1995)] is
considered. Starting from a state in which the vibrational modes are at a
temperature $T$, and each ion is in a superposition of an excited and a ground
state, an adiabatic approximation is used to find the inclusive probability
$P(t)$ for the ions to evolve as they would without the vibrations, and for the
vibrational modes to evolve into any final state. An analytic form is found for
$P(t)$ at $T=0$, and the decoherence time is found for all $T$. The decoherence
is found to be quite small, even for 1000 ions.
