A magnetohydrodynamic (MHD) stirrer that exhibits chaotic advection is investigated experimentally and theoretically. The stirrer consists of a circular cavity with an electrode (C) deposited around its periphery. Two additional electrodes (A) and (B) are deposited eccentrically inside the cavity on the bottom. The cavity is positioned in a uniform magnetic field that is parallel to the cylinder's axis, and it is filled with a weak electrolyte solution. Fluid motion is induced in the cavity by applying a potential difference across a pair of electrodes. A closed-form, analytical solution is derived for the MHD creeping flow field in the gap between the two eccentric cylinders. A singular solution is obtained for the special case when the size of the inner electrode shrinks to a point. Subsequently, passive tracers' trajectories are computed when the electric potential differences are applied alternately across electrodes AC and BC with period T. At small periods T, the flow is regular and periodic in most of the cavity. As the period increases, so does the complexity of the motion. At relatively large periods, the passive tracer experiences global chaotic advection. Such a device can serve as an efficient stirrer. Since this device has no moving parts, it is especially suitable for microfluidic applications. This is yet another practical example of a modulated, two-dimensional Stokes flow that exhibits chaotic advection.