Mechanical equilibrium at zero temperature does not necessarily imply thermodynamic equilibrium at finite temperature for a particle confined by a static, but non-conservative force field. Instead, the diffusing particle can enter into a steady state characterized by toroidal circulation in the probability flux, which we call a Brownian vortex. The circulatory bias in the particle's thermally-driven trajectory is not simply a deterministic response to the solenoidal component of the force, but rather reflects an interplay between advection and diffusion in which thermal fluctuations extract work from the non-conservative force field. As an example of this previously unrecognized class of stochastic machines, we consider a colloidal sphere diffusing in a conventional optical tweezer. We demonstrate both theoretically and experimentally that non-conservative optical forces bias the particle's fluctuations into toroidal vortexes whose circulation can reverse direction with temperature or laser power.