The weight of the body at the centre of earth is zero because at centre g value is zero.

$Weight = mg$

$g=\frac{GM}{R^2}$

$Mass = Volume \times Density$

$\mathrm{g}= \frac{\mathrm{G}{\displaystyle \frac{4{\mathrm{\pi R}}^3}{3}}\mathrm{\rho }}{{\mathrm{R}}^2}$

$\mathrm{g}= \frac{4\mathrm{G\rho \pi R}}{3}$

At the centre, the radius will be zero. Therefore, we can say the value of g will be zero.
Hence the weight will also be zero.