  We show that the $c=1$ bosonic string theory at finite temperature has two
matrix-model realizations related by a kind of duality transformation. The
first realization is the standard one given by the compactified matrix quantum
mechanics in the inverted oscillator potential. The second realization, which
we derive here, is given by the normal matrix model. Both matrix models exhibit
the Toda integrable structure and are associated with two dual cycles (a
compact and a non-compact one) of a complex curve with the topology of a sphere
with two punctures. The equivalence of the two matrix models holds for an
arbitrary tachyon perturbation and in all orders in the string coupling
constant.
