We show that the completely bound classical motion of a positive-energy electron is realized in the Coulomb field of a motionless nucleus and a uniform magnetic field. Such a motion exists due to conservation of the so-called invariant tori in the phase space of the system for not only the negative, but also for the positive energy of an electron. The completely bound trajectories occupy a much larger interval of the velocity directions compared with free trajectories for the same energy in a range of distances from the nucleus in which the typical time of the electron transit near the nucleus is larger than the cyclotron-gyration period, while the negative energy of Coulomb interaction is larger (in absolute value) than the total electron energy. The indicated range of distances is realized in the case of a low electron energy or a strong magnetic field when the Larmor radius of the electron is smaller than the characteristic impact parameter of the close Coulomb collisions in the absence of a magnetic field. The required conditions are realized in the photospheres of isolated magnetic white dwarfs and in the experiments on creation of antihydrogen.