Spin currents can apply useful torques in spintronic devices. The spin Hall effect has been proposed as a source of spin current, but its modest strength has limited its usefulness. We report a giant spin Hall effect (SHE) in β-tantalum that generates spin currents intense enough to induce efficient spin-torque switching of ferromagnets at room temperature. We quantify this SHE by three independent methods and demonstrate spin-torque switching of both out-of-plane and in-plane magnetized layers. We furthermore implement a three-terminal device that uses current passing through a tantalum-ferromagnet bilayer to switch a nanomagnet, with a magnetic tunnel junction for read-out. This simple, reliable, and efficient design may eliminate the main obstacles to the development of magnetic memory and nonvolatile spin logic technologies.