Using a numerical technique we demonstrate that the flow of the simplest differential viscoelastic fluid model (i.e., the upper-convected Maxwell model) goes through a bifurcation to a steady asymmetric state when flowing in a perfectly symmetric "cross-slot" geometry. We show that this asymmetry is purely elastic in nature and that the effect of inertia is a stabilizing one. Our results are in qualitative agreement with very recent experimental visualizations of a similar flow in the microfluidic apparatus of Arratia et al.