One possible means of counteraction against a hazardous asteroid is discussed: destruction of the object with a nuclear device during an earlier encounter with the Earth. This is feasible, since virtually all hazardous asteroids appear in near-Earth space several times before they hit the Earth. Computations show that this method is effective and essentially harmless if certain conditions are satisfied. Two possibilities are acceptable. In the first, a spacecraft overtakes the asteroid in a heliocentric orbit. In the second, the asteroid overtakes the spacecraft, which demands a substantially lower characteristic geocentric velocity for the spacecraft. This method for eliminating this cosmic threat is reasonable in two cases: when it is not possible to achieve a soft departure of the object from the collision orbit, and when the object continuously returns to the Earth. A soft departure from a collision orbit can be required multiple times, whereas the destruction of such an object must occur only once.