The temporal capillary instability of a viscoelastic liquid jet is studied by using weakly nonlinear stability analysis. The jet is supposed axisymmetric, of infinite length and evolving in an isotherm dynamically inert ambient medium. The liquid is considered incompressible and is represented by the Oldroyd-B rheological model. The free surface is assumed to be initially characterized by a single-mode deformation with a small amplitude compared to the radius of the undeformed jet. The analysis is performed up to second order using the small-amplitude perturbation method and a polynomial approximation of the terms containing products of modified Bessel functions of the first kind with different arguments as done in the recently studied Newtonian case (Renoult et al. 2018 J. Fluid Mech. 856:169-201). The temporal evolution of the surface shape, the velocity field, and the pressure field are then derived. These flow quantities depend on five dimensionless numbers: the dimensionless wavenumber, the dimensionless initial deformation amplitude, an Ohnesorge number and two Deborah numbers generated following the two time scales of the Oldroyd-B model: the stress relaxation and deformation retardation times. We denote that the Newtonian case can be attained by taking these two numbers equal. The spatio-temporal evolution of the jet will be shown for a large range of the five control parameters of the liquid mentioned above, and compared to the Niewtonian case.