We report a giant spin Hall effect (SHE) in {beta}-Ta that generates spin currents intense enough to induce efficient spin-transfer-torque switching of ferromagnets, thereby providing a new approach for controlling magnetic devices that can be superior to existing technologies. We quantify this SHE by three independent methods and demonstrate spin-torque (ST) switching of both out-of-plane and in-plane magnetized layers. We implement a three-terminal device that utilizes current passing through a low impedance Ta-ferromagnet bilayer to effect switching of a nanomagnet, with a higher-impedance magnetic tunnel junction for read-out. The efficiency and reliability of this device, together with its simplicity of fabrication, suggest that this three-terminal SHE-ST design can eliminate the main obstacles currently impeding the development of magnetic memory and non-volatile spin logic technologies.