We present a temporal stability analysis of a viscoelastic liquid jet. Pure-deformation initial conditions are imposed and the viscoelastic liquid is represented by an Oldroyd-B model. The analysis is performed up to second-order, with the small parameter being the dimensionless initial deformation amplitude of the imposed mode. The jet problem depends on four other dimensionless numbers: an Ohnesorge number, two Deborah numbers built on the characteristic times of the fluid model and the dimensionless wavenumber of the imposed mode. The results are compared to the Newtonian case, recently treated in the literature, that is retrieved by the identity of the two Deborah numbers. It is found that the first-order dispersion relation admits an additional solution that is associated to a decaying mode. Three behaviours are predicted according to the position of the wavenumber with respect to the cutoff wavenumber kc = 1, that is unchanged by viscoelasticity, and a critical-oscillation wavenumber k * that depends more on the Ohnesorge number than on the Deborah numbers: growth without oscillations for 0 < k ≤ k * , growth with oscillations for k * < k < kc and damping with oscillations for k ≥ kc. Concerning the second-order solution, it is found that the Poisson equation for the second-order pressure admits an additional contribution containing products of modified Bessel functions with different arguments, that requires a polynomial approximation to be solved. The second-order solution is obtained by following the same method as the one used for the Newtonian case, except that three modes need to be considered instead of two in the Newtonian case. Introduction Viscoelastic liquids are a common type of non-Newtonian fluids. These fluids behave like elastic solids at short times compared to the longest time of the fluid and like viscous liquids at longer times. Their elastic property, that is of scientific and industrial interest, is due to a macromolecular structure. Macomolecular fluids are challenging because their motions cannot be simply described by the Navier-Stokes equations. The diversity of their structures, the molecular weight distributions and the large number of internal degrees of freedom make their molecular modelling very different from the one of Newtonian liquids. This leads to a large number of molecular models in the literature [1]. The most common ones consist in representing the macromolecules by chains of N springs. In particular, the N=2 case, referred to as the elastic dumbbells, is very convenient to use as it minimises mathematical complications and is often sufficient to predict macroscopic behaviours. The theoretical description of a fluid flow is not only challenging when the fluid model is not Newtonian, but also when the geometry is not planar. These two difficulties have been put forward by Yarin in his seminal book [2]. In the case of a liquid jet, the cylindrical geometry adds a nonlinearity in the equations with respect to the spatial radial variable, generating Bessel functions in the solutions. The weakly nonlinear stability analysis of a Newtonian jet was recently treated by Renoult et al. [3] following the work of Yuen on the inviscid case [4]. Here, we propose to extend the study to the case of a Hookean dumbbells viscoelastic fluid. The assumptions and notations of the problem are first given. The problem is then formulated in dimensional and dimensionless equations. The method of solution is briefly recalled as it is the same as the one deployed for the Newtonian case [3]. The equations and solutions are finally presented for the two first orders of the analysis. The article ends with a conclusion.