The weight of the body is the force with which it is attracted towards the centre of the earth.

W = m × g

Weight of the body changes from place to place, as the value of g changes from place to place.

Acceleration due to gravity g is zero at the centre of earth, as at the centre of earth we are surrounded by equal masses in all the direction, hence equivalent gravitational force acting on us due to earth is zero and thus making acceleration due to gravity to zero.

So, W = m × g = 0

Hence, weight at the centre of earth is zero.