  All high-temperature phases of the known N=4 superstrings in five dimensions
can be described by the universal thermal potential of an effective
four-dimensional supergravity. This theory, in addition to three moduli s, t,
u, contains non-trivial winding modes that become massless in certain regions
of the thermal moduli space, triggering the instabilities at the Hagedorn
temperature. In this context, we look for exact domain wall solutions of first
order BPS equations. These solutions preserve half of the supersymmetries, in
contrast to the usual finite-temperature weak-coupling approximation, and as
such may constitute a new phase of finite-temperature superstrings. We present
exact solutions for the type-IIA and type-IIB theories and for a self-dual
hybrid type-II theory. While for the heterotic case the general solution cannot
be given in closed form, we still present a complete picture and a detailed
analysis of the behaviour around the weak and strong coupling limits and around
certain critical points. In all cases these BPS solutions have no instabilities
at any temperature. Finally, we address the physical meaning of the resulting
geometries within the contexts of supergravity and string theory.
