> An explanation is provided for the disruptive instability in diverted tokamaks when the safety factor at the 95 % poloidal flux surface, 95$]]> , is driven below 2.0. The instability is a resistive kink counterpart to the current-driven ideal mode that traditionally explained the corresponding disruption in limited cross-sections (Shafranov, Sov. Phys. Tech. Phys., vol. 15, 1970, p. 175) when edge$]]> , the safety factor at the outermost closed flux surface, lies just below a rational value . Experimentally, external kink modes are observed in limiter configurations as the current in a tokamak is ramped up and edge$]]> decreases through successive rational surfaces. For edge<2$]]> , the instability is always encountered and is highly disruptive. However, diverted plasmas, in which edge$]]> is formally infinite in the magnetohydrodynamic (MHD) model, have presented a longstanding difficulty since the theory would predict stability, yet, the disruptive limit occurs in practice when 95$]]> , reaches 2. It is shown from numerical calculations that a resistive kink mode is linearly destabilized by the rapidly increasing resistivity at the plasma edge when 95<2$]]> , but edge\gg 2$]]> . The resistive kink behaves much like the ideal kink with predominantly kink or interchange parity and no real sign of a tearing component. However, the growth rates scale with a fractional power of the resistivity near the surface. The results have a direct bearing on the conventional edge cutoff procedures used in most ideal MHD codes, as well as implications for ITER and for future reactor options.