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The gravitational spin-orbit interaction in the region of the ultrarelativistic orbital velocity of a spinning test particle in the Schwarzschild field is investigated on the basis of the Mathisson-Papapetrou equations. Different indications of an essential increase of this interaction, when the tangential velocity becomes ultrarelativistic, are analysed. The main factor of growth is the square of the relativistic Lorentz y-factor. Numerical estimations for a high-energy electron in its path near the Earth???s surface and near a neutron star are given. It is stressed that the widely recognised assumption that the deviation of a spinning test particle from the geodesic trajectory is caused by tidal forces is not correct. The real reason for this deviation is the gravitational spin-orbit interaction.