Materials with on-demand control of thermal conductivity are the prerequisites to build thermal conductivity switches, where the thermal conductivity can be turned ON and OFF. However, the ideal switch, while required to develop novel approaches to solid-state refrigeration, energy harvesting, and even phononic circuits, is still missing. It should consist of an active material only, be environment friendly, and operate near room temperature with a reversible, fast, and large switching ratio. Here, we first predict by ab initio electronic structure calculations that ferroelectric domains in barium titanate exhibit anisotropic thermal conductivities. We confirm this prediction by combining frequency-domain thermoreflectance and scanning thermal microscopy measurements on a single crystal of barium titanate. We then use this gained knowledge to propose a lead-free thermal conductivity switch without inactive material, operating reversibly with an electric field. At room temperature, we find a switching ratio of 1.6 $\pm$ 0.3, exceeding the performances of state-of-the-art materials suggested for thermal conductivity switches.