We measure the Bose-Einstein correlations (BEC) in
$\pi^0\pi^0$ pairs produced in Z$^0$ hadronic decays, using the 
full hadronic event sample collected at centre-of-mass energies at and 
near the Z$^0$ peak by the OPAL detector from 1991 to 1995. 
This corresponds to about 4 million hadronic $\rm Z^0$ decays. 
The measured correlation function is parametrised using 
a Gaussian shape, and the values for the strength and radius of the 
pion source are found to be compatible with those obtained for charged pions. 
The string model predicts a smaller source radius and a stronger BEC effect 
in $\pi^0\pi^0$ pairs than in charged pion pairs. However, these predictions 
hold for pairs produced directly from the string. 
According to our Monte Carlo simulations, the fraction of 
reconstructed $\pi^0\pi^0$ pairs originating from the string is
negligible, and so we have no sensitivity to test the string model     
predictions.