For the first time, the design and implementation of a fully-integrated wireless information and power transfer system, operating at 24 GHz and enabling battery-less sensor nodes, is presented in this paper. The system consists of an RF power source, a receiver antenna array, a rectifier, and a batteryless sensor node which communicates via backscatter modulation at 868 MHz. The rectifier circuits use commercially available Schottky diodes to convert the RF power to DC with a measured efficiency of up to 35%, an improvement of ten percentage points compared with previously reported results. The rectifiers and the receive antenna arrays were jointly designed and optimised, thereby reducing the overall circuit size. The battery-less sensor transmitted data to a base station realised as a GNU Radio flow running on a bladeRF Software Defined Radio module. The whole system was tested in free-space in laboratory conditions and was capable of providing sufficient energy to the sensor node in order to enable operation and wireless communication at a distance of 0.15 metres.