Electrons and other fundamental particles have an intrinsic angular momentum called spin. A change in the spin state of such a particle is therefore equivalent to a mechanical torque. This spin-induced torque is central to our understanding of experiments(1,2) ranging from the measurement of the angular momentum of photons(3) and the g-factor of metals(4-7) to magnetic resonance(8) and magnetization reversal in magnetic multilayers(8-15). When a spin-polarized current passes through a metallic nanowire in which one half is ferromagnetic and the other half is nonmagnetic, the spins of the itinerant electrons are 'flipped' at the interface between the two regions to produces a torque. Here, we report direct measurement of this mechanical torque in an integrated nanoscale torsion oscillator, and measurements of the itinerant electron spin polarization that could yield new information on the itinerancy of the d-band electrons. The unprecedented torque sensitivity of 1 x 10(-22) N-m Hz(-1/2) may have applications in spintronics and precision measurements of charge-parity-violating forces(16,17), and might also enable experiments on the untwisting of DNA(18) and torque-generating molecules(19,20).