A purely elastic flow instability in the recirculation flow of a driven half-cavity or blocked channel is discovered and examined both experimentally and numerically. Video image analysis is used to determine the critical Weissenberg number and characteristic wavenumber of the instability while laser sheet imaging provides information about the structure and location of the secondary flow. To understand the basic flow and ultimately the flow instabilities, we numerically solve the combined momentum conservation and constitutive equations describing the flow of. elastic fluids in the block flow geometry. Local rates of polymer stretch are estimated with a weakly elastic expansion using the Oldroyd-B constitutive equation for the polymer stress. Numerical solutions for the kinematics and polymer stretch are also obtained using the Chilcott-Rallison FENE model to describe the polymeric stress. These results are discussed in the context of known purely elastic instability mechanisms in order to better understand the stability of general recirculation flows.