It is shown experimentally that when an elastic liquid such as a polydimethylsiloxane is sheared in the gap of a cone-and-plate viscometer the liquid will fracture in shear at a certain critical stress. A previously published theory, based on the criterion that fracture will occur when the total elastic shear strain energy is such that a fraction of it is sufficient to supply the surface free energy of the new surface, correctly predicts the experimentally found dependence of the critical stress on the shape and volume of the sheared liquid. The critical stress increases with decreasing cone angle and cone radius, with decreasing separation of the cone apex from the plate, and with increasing volume of the meniscus of excess liquid surrounding the periphery of the cone. The theory has also been applied to the flow of an elastic liquid through a capillary and predicts that fracture should occur when a certain flow rate is exceeded. This rate should vary as the cube of the radius. Experimental results from the literature on the extrusion of polymers suggest that distortions of the extrudate, also known as melt fracture, which occur at definite flow rates are a manifestation of the predicted fracture.