In tokamak disruptions the Ohmic current is often replaced by a current of runaway electrons, which is likely to be more peaked in the center of the discharge than the predisruption current. This raises the question of the resistive stability of the postdisruption plasma, where the equilibrium current is entirely carried by the runaway electrons while the cold 10 eV background plasma is relatively resistive. It is found that the linear properties of the classical tearing mode are essentially determined by the cold bulk plasma, and the growth rate is approximately the same as in a plasma without runaways but with the same current profile. The nonlinear saturation amplitude is different, however. In a symmetric plasma slab, the saturated island size is larger when the current is carried by runaway electrons than in the Ohmic case, and undergoes a nonlinear bifurcation when the stability index exceeds a critical value.