  We consider a system in a non-equilibrium steady state by joining two
semi-infinite Ising chains coupled to thermal reservoirs with {\em different}
temperatures, $T$ and $T^{\prime}$. To compute the energy flux from the hot
bath through our system into the cold bath, we exploit Glauber heat-bath
dynamics to derive an exact equation for the two-spin correlations, which we
solve for $T^{\prime}=\infty$ and arbitrary $T$. We find that, in the
$T'=\infty$ sector, the in-flux occurs only at the first spin. In the
$T<\infty$ sector (sites $x=1,2,...$), the out-flux shows a non-trivial
profile: $F(x)$. Far from the junction of the two chains, $F(x)$ decays as
$e^{-x/\xi}$, where $\xi$ is twice the correlation length of the {\em
equilibrium} Ising chain. As $T\rightarrow 0$, this decay crosses over to a
power law ($x^{-3}$) and resembles a "critical" system. Simulations affirm our
analytic results.
