Extremely low-frequency magnetic fields are known to affect biological systems. In many cases, biological effects display "windows" in biologically effective parameters of the magnetic fields: most dramatic is the fact that the relatively intense magnetic fields sometimes do not cause appreciable effect, while smaller fields of the order of 10-100 microT do. Linear resonant physical processes do not explain the frequency windows in this case. Amplitude window phenomena suggest a nonlinear physical mechanism. Such a nonlinear mechanism has been proposed recently to explain those "windows." It considers the quantum-interference effects on the protein-bound substrate ions. Magnetic fields cause an interference of ion quantum states and change the probability of ion-protein dissociation. This ion-interference mechanism predicts specific magnetic-field frequency and amplitude windows within which the biological effects occur. It agrees with a lot of experiments. However, according to the mechanism, the lifetime Gamma(-1) of ion quantum states within a protein cavity should be of unrealistic value, more than 0.01 s for frequency band 10-100 Hz. In this paper, a biophysical mechanism has been proposed, which (i) retains the attractive features of the ion interference mechanism, i.e., predicts physical characteristics that might be experimentally examined and (ii) uses the principles of gyroscopic motion and removes the necessity to postulate large lifetimes. The mechanism considers the dynamics of the density matrix of the molecular groups, which are attached to the walls of protein cavities by two covalent bonds, i.e., molecular gyroscopes. Numerical computations have shown almost free rotations of the molecular gyroscopes. The relaxation time due to van der Waals forces was about 0.01 s for the cavity size of 28 A.