Recent investigations have somewhat clarified a supermolecular organization of polymeric fluids. Specifically, it has been found that at certain conditions water solutions of poly(ethylene oxide) (PEO) are two-phase fluids. As molecular weight increases one of two phases-the phase rich in polymer-first separates out as scattered globules. Then it builds up a liquid-crystalline fibrillar network. This network produces characteristic properties of PEO solutions: shear-thinning, elasticity, dynamic birefringence, macromolecular destruction. Most likely, similar properties of other polymeric fluids are generated in an analogous way. In this work for the same system, experiments have been pursued to find a surface that separates in the "intrinsic viscosity, concentration, temperature" space regions of the globular and the network state. Two methods were used. The first one is based on the fact that scission of PEO molecules stops at transition from the network state to the globular state. The second method employs the fact that only the network solutions possess shear-thinning properties. Previously published results are also used. The action of temperature was not detected. With increasing the intrinsic viscosity, the boundary concentration decreases. Data reveal that by dilution one can always obtain a solution in which PEO is not susceptible to destruction.