  We study the low-temperature transport properties of clean one-dimensional
spin-1/2 chains coupled to phonons. Due to the presence of approximate
conservation laws, the heat current decays very slowly giving rise to an
exponentially large heat conductivity, $\kappa ~ e^{T^*/T}$. As a result of an
interplay of Umklapp scattering and spinon-phonon coupling, the characteristic
energy scale $T^*$ turns out to be of order $\Theta_D/2$, where $\Theta_D$ is
the Debye energy, rather than the magnetic exchange interaction $J$ -- in
agreement with recent measurements in SrCuO compounds. A large magnetic field
strongly affects the heat transport by two distinct mechanisms. First, it
induces a LINEAR spinon--phonon coupling, which alters the nature of the $T ->
0$ fixed point: the elementary excitations of the system are COMPOSITE
SPINON-PHONON objects. Second, the change of the magnetization and the
corresponding change of the wave vector of the spinons strongly affects the way
in which various Umklapp processes can relax the heat current, leading to a
characteristic fractal--like spiky behavior of $\kappa$ when plotted as a
function of magnetization at fixed T.
