  Models are presented for CO rotational line emission by high redshift
starburst galaxies. The influence of the cosmic microwave background on the
thermal balance and the level populations of atomic and molecular species is
explicitly included. Predictions are made for the observability of starburst
galaxies through line and continuum emission between z=5 and z=30. It is found
that the Millimeter Array could detect a starburst galaxy with ~10^5 Orion
regions, corresponding to a star formation rate of about 30 Mo yr^{-1}, equally
well at z=5 or z=30 due to the increasing cosmic microwave background
temperature with redshift. Line emission is a potentially more powerful probe
than dust continuum emission of very high redshift objects.
