  Both the supersymmetric $SO(32)$ and $E_8\times E_8$ heterotic strings in ten
dimensions have known strong-coupling duals. However, it has not been known
whether there also exist strong-coupling duals for the non-supersymmetric
heterotic strings in ten dimensions. In this paper, we construct explicit
open-string duals for the circle-compactifications of several of these
non-supersymmetric theories, among them the tachyon-free $SO(16)\times SO(16)$
string. Our method involves the construction of heterotic and open-string
interpolating models that continuously connect non-supersymmetric strings to
supersymmetric strings. We find that our non-supersymmetric dual theories have
exactly the same massless spectra as their heterotic counterparts within a
certain range of our interpolations. We also develop a novel method for
analyzing the solitons of non-supersymmetric open-string theories, and find
that the solitons of our dual theories also agree with their heterotic
counterparts. These are therefore the first known examples of strong/weak
coupling duality relations between non-supersymmetric, tachyon-free string
theories. Finally, the existence of these strong-coupling duals allows us to
examine the non-perturbative stability of these strings, and we propose a phase
diagram for the behavior of these strings as a function of coupling and radius.
