We study experimentally the interfacial instability between a layer of dilute polymer solution and water flowing in a thin capillary. The use of microfluidic devices allows us to observe and quantify in great detail the features of the flow. At low velocities, the flow takes the form of a straight jet, while at high velocities, steady or advected wavy jets are produced. We demonstrate that the transition between these flow regimes is purely elastics, it is caused by the viscoelasticity of the polymer solution only. The linear stability analysis of the flow in the short-wave approximation supplemented with a kinematic criterion captures quantitatively the flow diagram. Surprisingly, unstable flows are observed for strong velocities, whereas convected flows are observed for low velocities. We demonstrate that this instability can be used to measure the rheological properties of dilute polymer solutions that are difficult to assess otherwise.