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A modal stability analysis shows that pressure-driven pipe flow of an Oldroyd-B fluid is linearly unstable to axisymmetric perturbations, in stark contrast to its Newtonian counterpart which is linearly stable at all Reynolds numbers. The dimensionless groups that govern stability are the Reynolds number, the elasticity number, and the ratio of solvent to solution viscosity. The unstable mode has a phase speed close to the base-state maximum over the entire unstable region in the relevant parameter space, implying that the unstable mode belongs to a class of viscoelastic center modes. Unlike the Newtonian transition which is dominated by nonlinear processes, the linear instability discussed here could be very relevant to the onset of turbulence in viscoelastic pipe flows. The prediction of an instability is, in fact, consistent with several experimental studies on pipe flow of polymer solutions, ranging from previous reports of early turbulence to the more recent discovery of elasto-inertial turbulence. The instability identified in this study comprehensively dispels the prevailing notion of pipe flow of viscoelastic fluids being linearly stable in the Reynolds-Weissenberg plane, marking a possible paradigm shift in our understanding of transition in rectilinear viscoelastic shearing flows.