  A system H with a Hagedorn-like mass spectrum imparts its unique temperature
T_H to any other system coupled to it. An H system radiates particles in
preexisting physical and chemical equilibrium. These particles form a saturated
vapor at temperature T_H. This coexistence describes a first order phase
transition. An H system is nearly indifferent to fragmentation into smaller H
systems. A lower mass cut-off in the spectrum does not significantly alter the
general picture. These properties of the Hagedorn thermostats naturally explain
a single value of hadronization temperature observed in elementary particle
collisions at high energies and lead to some experimental predictions.
