We implement a linear stability analysis for the viscous annular jet surrounded by inviscid motionless gas subject to three-dimensional disturbances, and an analytical dispersion relation is presented. With this relation, we are able to obtain evidence that the axisymmetric mode is the most unstable mode of the system. The evidence for general cases is based on the numerical results obtained by solving the transcendental dispersion relation for a wide range of physical parameters. The results for two special cases, a very thin annular jet and an annular jet with disturbances of very short axial wavelength, are obtained from two rigorous analytical approaches.