  Fundamental conservation laws predict ballistic, i.e., dissipationless
transport behaviour in one-dimensional quantum magnets. Experimental evidence,
however, for such anomalous transport has been lacking ever since. Here we
provide experimental evidence for ballistic heat transport in a S=1/2
Heisenberg chain. In particular, we investigate high purity samples of the
chain cuprate SrCuO2 and observe a huge magnetic heat conductivity
$\kappa_{mag}$. An extremely large spinon mean free path of more than a
micrometer demonstrates that $\kappa_{mag}$ is only limited by extrinsic
scattering processes which is a clear signature of ballistic transport in the
underlying spin model.
