  Thermal conductivity (\kappa) of a distorted spin diamond-chain system,
Cu_3(CO_3)_2(OH)_2, is studied at low temperatures down to 0.3 K and in
magnetic fields up to 14 T. In zero field, the \kappa(T) curve with heat
current along the chain direction has very small magnitudes and shows a
pronounced three-peak structure. The magnetic fields along and perpendicular to
the chains change the \kappa strongly in a way having good correspondence to
the changes of magnetic specific heat in fields. The data analysis based on the
Debye model for phononic thermal conductivity indicates that the heat transport
is due to phonons and the three-peak structure is caused by two resonant
scattering processes by the magnetic excitations. In particular, the spin
excitations of the chain subsystem are strongly scattering phonons rather than
transporting heat.
