The problem to determine the location and shape of a perfect conductor within a conducting homogeneous host medium from measured current and voltages on the accessible exterior boundary of the host medium can be modelled by an inverse Dirichlet boundary value problem for the Laplace equation. For this, recently Kress and Rundell suggested a novel inverse algorithm based on nonlinear integral equations arising from the reciprocity gap principle. The present paper extends this approach to the problem to recover the location and shape of a rigid body immersed in a fluid from the measured velocity and traction at the exterior boundary of the fluid, that is, to an inverse boundary value problem for the Stokes equation. The mathematical foundation of this extension is provided and numerical examples illustrate the feasibility of the method.