  A current of magnetic moments will flow in the spin-1/2 Heisenberg chain in
the presence of an external magnetic field $B$ and a temperature gradient
$\Delta T$ along the chain. We show that this magnetothermal effect is strictly
{\em infinite} for the integrable Heisenberg-model in one dimension. We set-up
the response formalism and derive several new generalized Einstein relations
for this magnetothermal effect which vanishes in the absence of an external
magnetic field. We estimate the size of the magnetothermal response by exact
diagonalization and Quantum Monte Carlo and make contact with recent transport
measurements for the one-dimensional Heisenberg compound $\rm Sr_2CuO_3$.
