  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.
