  The Hagedorn transition in string theory is normally associated with an
exponentially rising density of states, or equivalently with the existence of a
thermal string winding mode which becomes tachyonic above a specific
temperature. However, the details of the Hagedorn transition turn out to depend
critically on the precise manner in which a zero-temperature string theory is
extrapolated to finite temperature. In this paper, we argue that for broad
classes of closed string theories, the traditional Hagedorn transition is
completely absent when the correct extrapolation is used. However, we also
argue that there is an alternative ``re-identified'' Hagedorn transition which
is triggered by the thermal winding excitations of a different, ``effective''
tachyonic string ground state. These arguments allow us to re-identify the
Hagedorn temperature for heterotic strings. Moreover, we find that all
tachyon-free closed string models in ten dimensions share the same (revised)
Hagedorn temperature, resulting in a universal Hagedorn temperature for both
Type II and heterotic strings. We also comment on the possibility of thermal
spin-statistics violations at the Planck scale.
