Analysis and experiments have shown that the flow of a viscoelastic fluid between a rotating cone and plate is unstable to a three-dimensional time-dependent instability which results in a secondary motion that, close to onset, has the form of a Bernoulli spiral. Here we report results of a linear stability analysis for a multi-mode formulation of the Giesekus constitutive equation with parameters determined by regression to the linear relaxation spectrum and steady-state shear flow properties of the constant viscosity solution of polyisobutylene, polybutene and tetradecane first characterized by Quinzani et al. (J. Rheol., 34(1990) 705-748). The numerically calculated stability boundaries for the onset of the elastic instability are compared to flow visualization experiments and provide quantitative agreement for both the critical Deborah number and the non-axisymmetric azimuthal structure of the spiral instability. The results are qualitatively similar to the predictions obtained from non-linear models with a single relaxation time, such as a single-mode Giesekus model or the FENE dumbbell model proposed by Chilcott and Rallison; however, the more precise fit of the first normal stress coefficient supplied by the multi-mode model appears necessary for quantitative prediction of the experiments. Quasi-linear models lacking shear-thinning behavior in the first normal stress coefficient Psi(1) (gamma), such as single- or multi-mode formulations of the Oldroyd-B model, are incapable of even qualitative prediction of the dependence of the stability boundaries on Debroah number observed in experiments with viscoelastic polymer solutions.