  We describe finite temperature N=4 superstrings in D=5 by an effective
four-dimensional supergravity of the thermal winding modes that can become
tachyonic and trigger the instabilities at the Hagedorn temperature. Using a
domain-wall ansatz, exact solutions to special BPS-type first order equations
are found. They preserve half of the supersymmetries, contrary to the standard
perturbative superstring at finite temperature that breaks all supersymmetries.
Our solutions show no indication of any tachyonic instability and provide
evidence for a new BPS phase of finite temperature superstrings that is stable
for all temperatures. This would have important consequences for a stringy
description of the early universe.
