The stability of an axisymmetric jet was examined in the presence of external co-flow and counterflow. Spatio-temporal theory was used to distinguish regions of absolute and convective instability in a parameter space including the velocity ratio, density ratio, Mach number, and the shear layer thickness. The absolute-convective transition was identified for two distinct axisymmetric modes. One of these modes became absolutely unstable in the presence of ambient co-flow while the other mode required external counterflow to admit an absolutely unstable solution. In general, the former mode was most unstable in low density jets, while the latter became more unstable as the jet density was increased relative to the surrounding fluid. In the range of parameters studied, both modes became increasingly unstable with decreasing jet density and for lower Mach numbers. The results of the spatiotemporal theory are also compared to globally unstable modes identified in laboratory jets.