  The heat transfer model for a one-dimensional supercooled melt during the
final stage of solidification is considered. The Stefan problem for the
determination of the temperature distribution is solved under the condition
that (i) the interface approaches the specimen surface with a constant velocity
$V$; (ii) the latent heat of solidification linearly depends on the interface
temperature; (iii) all the physical quantities given at the phase boundary are
presented by linear combinations of the exponential functions of the interface
position. First we find the solution of the corresponding hyperbolic Stefan
problem within the framework of which the heat transfer is described by the
telegraph equation. The solution of the initial parabolic Stefan problem is
then found as a result of the limiting transition $V/V_H \rightarrow 0$ $(V_H
\rightarrow \infty)$, where $ V_H $ is the velocity of the propagation of the
heat disturbances, in which the hyperbolic heat model teds to the parabolic
one.
