We consider a stochastic process of heat conduction where energy is redistributed along a chain between nearest neighbor sites via an improper beta distribution. Similar to the well-known Kipnis-Marchioro-Presutti (KMP) model, the finite chain is coupled at its ends with two reservoirs that break the conservation of energy when working at different temperatures. At variance with KMP, the model considered here is integrable and one can write in a closed form the $n$-point correlation functions of the non-equilibrium steady state. As a consequence of the exact solution one can directly prove that the system is in a `local equilibrium' and described at the macro-scale by a product measure. Integrability manifests itself through the description of the model via the open Heisenberg chain with non-compact spins. The algebraic formulation of the model allows to interpret its duality relation with a purely absorbing particle system as a change of representation.