The maximum strength of toroidal magnetic field in the Earth's core is estimated on the basis of the stability of the magnetic field. In our model we take our basic magnetic field B0 to be composed of both toroidal and poloidal axisymmetric decay modes of lowest order. While the strength of the poloidal component, BP is taken consistent with observation, the maximum strength of the toroidal field, ‖BT‖max, is regarded as a parameter of the model. By demonstrating that viscous dissipation is of secondary importance and therefore that the results are independent of the parameter associated with viscosity, our model is eventually dependent on only one parameter: the ratio A of the maximum strength of the invisible toroidal field to the strength of the poloidal field at the pole of the core-mantle boundary. It is shown that ‖BT‖max≤10‖BP(&thgr;=0,r=r0)‖ in order that the basic magnetic field B0=BP+BT is stable, giving an estimated upper bound on strength of the invisible toroidal field of order 50 gauss. ¿ American Geophysical Union 1993