  We report calculations designed to assess the ultimate precision of an atomic
clock based on the 578 nm $6 ^1S_0 --> 6 ^3P^o_0$ transition in Yb atoms
confined in an optical lattice trap. We find that this transition has a natural
linewidth less than 10 mHz in the odd Yb isotopes, caused by hyperfine
coupling. The shift in this transition due to the trapping light acting through
the lowest order AC polarizability is found to become zero at the magic trap
wavelength of about 752 nm. The effects of Rayleigh scattering, higher-order
polarizabilities, vector polarizability, and hyperfine induced electronic
magnetic moments can all be held below a mHz (about a part in 10^{18}), except
in the case of the hyperpolarizability larger shifts due to nearly resonant
terms cannot be ruled out without an accurate measurement of the magic
wavelength.
