We report experimental observation of a localized structure, which is of a new type for dissipative systems. It appears as a solitary vortex pair (``diwhirl'') in Couette flow with highly elastic polymer solutions. In contrast to the usual solitons the diwhirls are stationary. It is also a new object in fluid dynamics-a pair of vortices that build a single entity. The diwhirls arise as a result of a purely elastic instability through a hysteretic transition at negligible Reynolds numbers. It is suggested that the vortex flow is driven by the same forces that cause the Weissenberg effect.