  We consider structure of a thermal phase-slip center for a simple microscopic
model of a clean one-dimensional superconductors in which superconductivity
occurs only within one conducting channel or several identical channels.
Surprisingly, the Eilenberger equations describing the saddle-point
configuration allow for exact analytical solution in the whole temperature and
current range. This solution allows us to derive a closed expression for the
free-energy barrier, which we use to compute its temperature and current
dependences.
