Elementary excitations in the spin-ice compound Dy$_2$Ti$_2$O$_7$ can be described as magnetic monopoles propagating independently within the pyrochlore lattice formed by magnetic Dy ions. We studied the magnetic-field dependence of the thermal conductivity {kappa}(B) for B || [001] and observe clear evidence for magnetic heat transport originating from the monopole excitations. The magnetic contribution {kappa}_{mag} is strongly field-dependent and correlates with the magnetization M(B). The diffusion coefficient obtained from the ratio of {kappa}_{mag} and the magnetic specific heat is strongly enhanced below 1 K indicating a high mobility of the monopole excitations in the spin-ice state.
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