At second order in perturbation theory, the unstable r -mode of a rotating star includes growing differential rotation whose form and growth rate are determined by gravitational-radiation reaction. With no magnetic field, the angular velocity of a fluid element grows exponentially until the mode reaches its nonlinear saturation amplitude and remains nonzero after saturation. With a background magnetic field, the differential rotation winds up and amplifies the field, and previous work where large mode amplitudes were considered [L. Rezzolla, F. K. Lamb, and S. L. Shapiro, Astrophys. J. 531, L139 (2000)., 10.1086/312539], suggests that the amplification may damp out the instability. A background magnetic field, however, turns the saturated time-independent perturbations corresponding to adding differential rotation into perturbations whose characteristic frequencies are of order the Alfvén frequency. As found in previous studies, we argue that magnetic-field growth is sharply limited by the saturation amplitude of an unstable mode. In contrast to previous work, however, we show that if the amplitude is small, i.e., ≲10-4 , then the limit on the magnetic-field growth is stringent enough to prevent the loss of energy to the magnetic field from damping or significantly altering an unstable r -mode in nascent neutron stars with normal interiors and in cold stars whose interiors are type II superconductors. We show this result first for a toy model, and we then obtain an analogous upper limit on magnetic-field growth using a more realistic model of a rotating neutron star. Our analysis depends on the assumption that there are no marginally unstable perturbations, and this may not hold when differential rotation leads to a magnetorotational instability.