Spin transistors and spin Hall effects have been two separate leading directions of research in semiconductor spintronics which seeks new paradigms for information processing technologies. We have brought the two directions together to realize an all-semiconductor spin Hall effect transistor. Our scheme circumvents semiconductor-ferromagnet interface problems of the original Datta-Das spin transistor concept and demonstrates the utility of the spin Hall effects in microelectronics. The devices use diffusive transport and operate without electrical current, i.e., without Joule heating in the active part of the transistor. We demonstrate a spin AND logic function in a semiconductor channel with two gates. Our experimental study is complemented by numerical Monte Carlo simulations of spin-diffusion through the transistor channel.