Possible paths for obtaining linear stability against the m=0 mode in the Z-pinch are studied. Using a generalized energy principle, the necessary and sufficient Chew-Goldberger-Low (CGL) m=0 stability criterion is derived. This criterion is less restrictive than that of ideal MHD, although it also requires the boundary plasma pressure to be finite. It is shown that the edge pressure cannot be stably upheld by a surface current. By instead assuming a finite pressure external gas, it is found that an edge pressure to on-axis pressure ratio of 0.5 is required for stability of a constant current density profile. A parabolic current density profile lowers the limit to the value 0.17. The growth rates are shown to be monotonically decreasing as a function of the external gas pressure. Detailed derivations of the boundary conditions are also given. The results aid in clarifying the experimental stability of four major Z-pinch experiments. Finite Larmor radius stabilization is hence required to maintain stability in future fibre pinch experiments in vacuum, implying line densities less than 1019 m-1